Now, for the rest of our day, we're going to sit in section 4.6. We're going to talk about section 4.7, which has to do with counting. So, are you ready?
I'll start. One, two, three. Oh, we have a lot of work to do.
Ready? One, two, three. No, that's not the counting.
I'm sorry. Yeah, we're not doing that type of counting. That would be easy. That would be nice.
We just count it one, two, three, four, five. What we're doing in this type of counting is counting the ways that something can happen. That's what this is called counting. Counting the ways that some group of items can be structured, reordered, how many ways things can occur. We're going to deal with words called permutations and combinations in a little while.
The first thing we're going to talk about, though, is something called the fundamental count. counting rule and it has to do with this. It says let's say now that I have a coin and a die.
We are going to count in this section. How many ways procedures can happen? My procedure is I'm going to flip the coin, then roll the die.
That's what's going to happen. And we're going to count up how many ways we can, how many combinations of outcomes we have in this case. So let's look at the coin.
How many ways can you flip the coin, how many outcomes can you get when you flip the coin? You really can only flip the coin like one way, right? You go like that. But how many outcomes can you get when you flip the coin? Two outcomes.
How many ways? Okay, how many outcomes can you get when you roll a die? How many outcomes can you get together when you flip a coin and roll a die?
Say that. Someone said I heard it. I thought I heard it. Isn't that wrong?
What now? Okay. Let's do this a different way.
There's how many outcomes for a quorum? What are the outcomes? There's how many outcomes for the die? What are the outcomes?
So if you get a head, could you get a 1, 2, 3, 4, 5, or 6? If you get a tail, could you still get a 1, 2, 3, 4, 5, or 6? So let me ask this question again. How many total outcomes do you get?
You get head 1, head 2, head 3, head 4, head 5, head 6. How many is that? Tail one, tail two, tail three. All together you get your 12 outcomes.
Do you see your 12 outcomes? How are you going to get four from here to 12? Here's what the fundamental counting rule says. Fundamental counting rule says this. If you have one event that can occur M ways and you have another event that can occur N ways, then together they can occur M times N ways.
Here's why. Every outcome you have for the coin, you have this many outcomes for the die. Does that make sense?
Every different outcome here, you still have six outcomes. If I had a coin with three sides somehow, I'd have three times six, because every different change of this outcome means you still have six outcomes here. You with me on this, Fundamundi County rule? That's what the Fundamundi County rule says.
It says that if you have an event that can occur m ways, Thank you. And another event that can occur n ways. Together They can occur m times n ways. And the reason is, again, if you have m ways this can occur, for every way, your second event can still occur n ways.
So we multiply every outcome here times every outcome here. That's why we get the m times n. Together they can occur m times n ways.
What's kind of cool is this concept can be extended indefinitely, which is nice. Let me show you an example of that and then we'll be done for today. Have you ever forgotten your pin number for your ATM? Have you?
How many people know their pin number for the ATM? What is it? Okay. Just kidding.
Just kidding. I would never do that to you unless I really needed the money. This concept can be extended. Let's say that you forgot your pin number. Pin numbers are usually how many digits for you guys?
Okay, good. So for a security code. Security code, let's say that you need to make up or you have forgotten your pin number. You have no idea what it is.
All you remember is that when you made it up, the first digit could not be zero because typically you can't have a zero on your first digit. It's like 1000 could be the first one you can get. So with your security code, we know the first digit cannot be zero. What I want to know is how many pin numbers do you have to check or how many pin numbers are possible when you're creating this?
How many would you need to check if you completely forgot it? I mean, potentially check. Let's see. If the first digit can't be zero, how many options do you have for the first number?
One, nine, ten. It can't be zero. Digits are zero, one, two, three, four, five, six, seven, eight, nine. Okay? So, ten is not a digit because it has more than one digit.
So, if you can't have zero as your first digit, how many options do you have for the first one? You have nine. You don't have zero, but you have one, two, three, four, five, six, seven, eight, nine. Are you with me? You have nine.
Now, how many choices do you have for the second digit? Nine and ten. Can you pick the zero? Yeah.
Sure you can. It is only for the first digit you can't. So you have zero, one, two, three, four, five, six, seven, eight, nine.
That's how many, that's ten digits. How many for the third one? Ten. Oh, that's close. There is no stipulation on that.
I almost got nailed to the pen. How about for the last one? Scared me. This is a fundamental counting rule problem. Here's why.
For every choice, of your first digit, you have 10 choices for the second, 10 choices for the third, and 10 choices for the fourth. Does that make sense to you? So every time you choose here, you have that many more choices. If I multiply those all together, that's how many total options I have for pin numbers.
How many is that? 9,000 total combinations. That means that if you, hey, let's extend this concept in our last minute here. If we have 9,000 total combinations and you're trying to break into someone's ATM, I know what you guys do on the weekends. You don't fly here.
Just kidding. If you're trying to break into someone's ATM and you get one shot at it, what's your probability you're going to get it right? One out of how many?
One shot? Out of 9,000, what would be the probability that you get it right on the first guess? Probability of guessing with one try.
However, if you try again, that probability is doubled. Well, not doubled, I'm sorry. That probability goes up. Not necessarily doubled, I suppose.
But if you try again, second try, that probability, if you try a different number, I mean, hopefully you don't try the same number several times. You'd be the worst snake ever. But if you tried like a hundred times, you'd have, well, one out of 900, right?
That's significantly better. Or if you try 1,000, eventually you're gonna get this, right, there's only 9,000 combinations, actually not that many. But your probability's still pretty low if you guess it once. But do you understand how the combination idea works?
and how the fundamental counting rule can be applied towards our probability. We'll feel okay with that so far. We're gonna try a lot more examples next time, but we're gonna end next day.
So we're talking about the fundamental counting rule and we're thinking that if we have a certain number of choices for each of these events, and now in our case for a security code, picking a digit is an event. So what we have here is an extension of the fundamental counting rule. It says you have an event of picking a digit.
Then you have this event of picking a digit. Then another event of picking a digit. And so on and so forth, so you get five digits. Those are all separate events in this context.
So what we're doing is figuring out how many ways can you pick the first digit? How many ways can you pick the second and the third and the fourth and the fifth? Because for every choice you have here, you have a different number of choices for each one of these.
For every choice you have here and here, you have a different number of choices. That becomes a fundamental counting rule problem. That means we're going to multiply. each of those different choices to find out how many total security codes we have for a five digit number.
Are you with me on this? Yes or no? Yes.
Are you sure? Yes. So let's see how many choices we have for the first digit.
If the first digit can't be zero and it can't be one, what that leaves you with are the digits 2, 3, 4, 5, 6, 7, 8, or 9. That's eight choices there. Did you get eight choices for the first one? So we got eight choices for...
the first digit. Here's what I'm talking about. For each of the eight choices you have here, you could pick a number for this, couldn't you?
So if you pick one, I'm sorry, you can't pick one, if you pick two, let's say the number was two. Let's say you pick the number two. You could pick a zero, a one, a two, a three, a four, a five, a six, a seven, an eight, or a nine for that, couldn't you? That's this many eight times this many choices. For every choice you have here, you can pick those.
numbers if I chose this to be a five I could still pick a zero a one the two all the way through nine for this one if I think that to be a nine I can pick a zero through nine for that one so for every choice you pick here you have another set of choices here how many do you have to pick from in that good because I made no stipulations I said you couldn't be zero or one that means eight choices but here you certainly have ten choices to choose from how about for a third digit how many Yeah. Fourth digit? Nine.
Fifth digit? Nine. Why nine?
Because it's right there. Okay. If we multiply those things together, that's going to tell us how many possible security codes we have with the five-digit number. How much is that? 72. That's right.
The fundamental accounting rule says if you have a certain number of, an event that can occur a certain number of ways, for instance we have five different events here, to find out the ways they can occur together, you multiply them. So we have 72,000 different codes. How many did we have with the four?
Four digit code. Yeah, and I only made one limitation. It said the first digit couldn't be zero. We had 9,000. So just by adding that extra little number, look how much more choices you have.
That's crazy, right? That's a lot more choices. much more secure code.
There's only 9,000 choices for a four-digit PIN, your ATM code. So if you punched in 9,000 different ones, unless they count to zero, then of course it would just be 10,000. But if you add the extra digit on there, it's much more secure, lots more choices, so someone can't randomly just pick it.
How many of you understood this problem? Are you okay with it? Good.
Let's go ahead and take this a different way. Let's say that you have... You have five different things. You've got apple.
You've got orange. That's an orange. You've got banana.
Grapes. What's another fruit missing? Oh, kiwis! You get kiwis.
Everyone likes kiwis. Who said that? I feel so sorry for you. It's the best fruit ever invented. Does it make your mouth all bubbly and stuff?
It makes my throat all swollen. deal with it. I think they're so acidic, like it gives me bubbles on my tongue, but I eat them anyway, I don't care. They're so good.
Give these five fruits. It could be five of anything really that are different. What I'm asking is, how many How many ways can we put these five fruits in order? So how many different arrangements can we get out of this? One, two, three, four.
There's five different spaces. How many ways can we arrange these fruits? So this is one arrangement, right? We do what I call these things? What's that?
Apple, orange, banana, grapes, kiwi. We could do it that way. We could switch these two.
That would be another way. We could switch these two. That would be another way. We could put this one first. That would be another way.
We could do all that and write them all out. This is going to take us forever, right? We're going to draw a whole bunch of fruit by the end of this, and we're not going to want to do that.
However, if we think about it like a fundamental counting rule, similar to that idea, how many choices would you have to put in your first order? So you're going to line these up on a shelf or something, or you're going to eat these in a certain order. You're going to eat a whole lot of fruit today.
How many choices do you have for your first consumption? Five. Yeah, there's five fruits. You haven't eaten any of them yet.
So there's five. choices that you could eat first but as soon as you eat the fruit it's gone how many choices do you have to eat second? Four. Then you eat the second fruit it's gone how many choices you have for the third one?
Three. And then? Two.
And then after you eat all these fruits you only have one fruit left there's only one choice at the very end and then well you're done you're out of fruit you don't have anything else you can replace there. Does this make sense to you, why our choices drop each time? So if we're putting things in order that are different, that are distinct or unique, we do have however many we have, that's how many choices we have for the first one, but as soon as you place that in the first spot, your choices drop by one.
one, then drop by one. Every time you choose one of those unique items, you have a different, a lower choice for the second one. Yes? So what would that be if you're not taking it out?
Like if you're not eating it, or if you're not, it's still there, but it's just a different order. Right, we're just counting the different orderings we could have this. Why does it take one away every time?
You're eating it. Or if you're not eating it, what we're doing is we're counting the number of arrangements you could have. So this would count as one. one ordering, right?
One ordering. As soon as I place this somewhere, as soon as I place one of these fruits somewhere, like at the beginning, I only have four choices to place a second fruit. It's not 5-5-5 because I don't have five of these different fruits all the time. I just have one apple, and I have one orange, one banana, one set of grapes, and one kiwi. So as soon as I place one down here, I say I have five choices of the first pick.
I would say picking a team or something. or something. I have five choices for the first pick, but as soon as I pick that one to get the rest of my arrangement, I only have four choices for the next spot.
Then I only have three choices for the next spot. Then only two choices for the next spot, but finally I'm left with one. You can do this any way you want. If I rearrange these, that's what we're counting is the number of arrangements we can make out of this. The number of different combinations or I've used the word combinations, actually called a permutation, which we'll talk about in a second.
Different arrangements. of these five fruits. If you don't want to consider them like you're actually eating them, because we don't want them to go away or anything, you could say, how many pictures could I take that were different of these fruits? How many different pictures?
If I don't exchange anything, that's one picture of the ordering of this fruit. Does that make sense? Then if I change these two, would that be a different looking picture to you?
If I change these two, would that be a different looking picture to you? to you. This is how many arrangements we have that would give you different pictures. So if you took your snapshot out and just take a look at those, this would be one way to do it. We have five choices for our first spot for our fruits.
After that, that one is set. You only have four choices for the next one, then three, then two, then one. You take away that choice after you put this in a certain order.
So for instance, let's make one more arrangement. I can't draw these all day. Let's make one more arrangement. Let's say I pick banana first.
You with me? How many did I have to choose from? Five. I could have picked any one of those first, right? That's the banana first.
Now that I've picked a banana, can I write another banana? I don't have another banana to choose from, unfortunately, because I like bananas. Still, they give me the bumps in my tongues, too. I think I'm just allergic to fruit. But as soon as I pick the banana, I have how many choices for the next one?
Okay, so what's the next choice that you want to pick? No, that's the wrong choice. I'm just kidding.
Yeah, orange. I have the banana, I have the orange. How many choices do I have for the next one?
Okay, what do you want to pick? Can I pick a banana and the orange again? That wouldn't make sense. So, I heard kiwi.
Kiwi. It's like a hairy baby. That's what it looks like to me. Hairy baby face. Okay, so we have hairy baby face.
What's the next one? How many choices do I have for the next one? Doesn't it look like a hairy baby face?
Yeah, it looks like a little hairy baby monster face. Anyway, that's a kiwi. So after I pick banana, then orange, then kiwi, I do... I only have two choices, though.
Do you get it? I can only choose between my apple or my grapes. That's the only thing I can choose from. I'll choose apple.
How many choices do I have for the last one? There's only one thing left. There's just the grapes.
That's why you go down one for every single arrangement that you, for every single choice that you have. So you had five choices here but then four, three, two, one. This counts as a different arrangement than this one.
Does that make sense? The number of ways we can do this is I had five choices first, four choices second, three, two, then one. You multiply all those together, you're going to get how much?
Did they have to be fruits? Could I have picked books? Five different books?
Could have done that, right? Or five different coins? Or five different colored marbles or something like that? No matter what, as long as they're unique.
unique items, what we do to find out how many arrangements we can get out of that, you multiply them. But your choice just drops every time. You pick one, the choices drop down.
You pick another one, the choices drop down. So this gives us 120 different ranges of five unique items. I'm not gonna say fruits because this could go for any, any item, any five items that happen to be different.
This could work for people. If I pick five different people and I put you in a line, I have five choices for the first person, but then I can't put that first person twice, can I? That wouldn't even make sense. I can't put the same person five times in a row. I have five choices for the first person in line, then I have only four choices for the second person, and three and then two, then one.
But we can represent this a little bit more nicely. What you need to know is that this, Five times four times three times two times one. That's actually a mathematical thing.
Have you heard of it before? Have you seen it? You might have seen it before.
Have you ever had a class similar to this one? We can write this as five and we use a nice symbol after it. It's an exclamation point. It's five factorial.
It's not five, okay? I'm so excited about that five. It's not like five. No, it's five factorial.
In mathematics, that exclamation point means this idea. Did I just scare you? I saw you kind of. I told you I was going to wake you up today.
This says it's a factorial. Doesn't mean you're super excited about the math, OK? Even though it's awesome.
Factorial. Did I scare you? Yeah. So the 5 times 4 times 3 times 2 times 1, instead of writing all of that out, we can say 5 factorial.
When you see the factorial, what it means is whatever number you have, sequentially, be multiplied sequentially by its successive numbers downwards until you get to 0. Okay, so 5 factorial means 5 times 4 times 3 times 2 times 1. 10 factorial would be 10 times 9. 9 times 8 times 7 times 6, all the way down to get to 1. The 0, we have a definite definition for that. 0 factorial is defined to be 1. That's just a definition. We have to have that, otherwise the factorials do not work. So here's what this tells us.
This isn't just about fruit here. It's not just about colored marbles. This is about the unique items.
That's really what I wanted to get across to you. Or five unique people, or anything like that. This is about the arrangement of unique items. Here's what we learned. If I had...
of 10 different fruits. I would have 10 choices for the first one, then nine, then eight, then seven, then six, all the way down to one. Are you with me on this, folks? So what the arrangement of these unique items gives us is that any n unique items can be arranged n factorial different ways.
For any set of n different items, The key word there is different, though. Another word that I used earlier was unique, different or unique. For any set of n different items, there are n factorial different arrangements that you can make of it.
For any set of n different items, there are n factorial different arrangements possible. All this says, says shoot, if there was just more than 5 fruits, if there was like 9, you could arrange up 9 times 8 times 7 times 6 times 5. That'd be 9 factorial. If there was a hundred different items, a hundred different people, how many ways did you put them in order?
A hundred choices from the first one. the 98, the 97, the 96. It's just extending this concept. We can just say it this way, though.
For n different items, in this case we had five. There was five factorial ways we could arrange it. For any different items, there's n factorial different ways we can arrange it. Raise your hand if you feel OK about that, understand it. Good.
You ever been to Disneyland? There's only seven good rides at Disneyland, actually. There's the one that they blow the smells in your face.
You know that one? Yeah, that one. I just did like this.
Universal for sorry. There's that one. There's the loop. There's a teacups, love the teacups.
Even though you throw up on them while they're so good. Have a churro and a hamburger, boom, guaranteed vomit cup. That's what that is.
That's why they're in a cup. It's right there. Then there's Space Mountain, and there's the, don't they have a dropy one?
Dropy? The dropy one, yeah. The Haunted Hotel?
Yes, the Haunted Hotel, that's pretty cool. There's two other ones. Oh, the Buzz Lightyear, that's kind of lame, but whatever, we'll count that. And then...
Indiana Jones. Yeah, the line's like 14 hours long, oh my gosh. You get the Fast Pass. That's a jip, you only get one at a time, or two at a time.
Okay, Indiana Jones, whatever, you're pissing my off. There's only seven good... rides at Disneyland. Now you're going to go early in the morning, right? And you're going to get there, so you're going to be the first person on those rides, but you've got to determine which ride you're going to go on first.
So we're going to have a game plan. We're going to say we're going to go to Indiana Jones first, because that's the longest one. Then we're going to go to like Space Mountain, then we're going to do this, then that, then that, then two other ones, and we'll call it a day.
Does that make sense? How many ways could we do that? How many different orderings could you have of those seven rides?
There are seven cool rides at Disneyland. How many different ways could you ride all seven? Well, let's think about it. Are the... Are the rides unique?
Are they distinct? Yeah, Indiana Jones is not the same as Space Mountain. Those are seven distinct rides that you're going on. How many choices would you have for your first ride?
Seven. Then after you win on that one, you only have... Six.
And then so on. So this, since these are n or 7 distinct different rides, what our factorial says up here is that we have 7 factorial different ways we can ride those. 7 for the first choice, 6 for the next choice, then 5, then 4, then 3, then 2. Then after you exhaust all your resources, you only have one ride you can go on at the very end.
So that's how we're getting the 7 factorial. How many ways is that? How much is that?
Probably need to use a calculator on that. How much? 5,040.
5,040? Wow. Wow. So you did 7 times 6 times 5 times 4 times 3 times 2 times 1, right? But you really don't have to do the times 1. You know, we know that one.
But 5,040. So you could go, see, there's only 7 good rides. You could have 5,040 different experiences at Disneyland.
Isn't that awesome? Just a different order. It would give you a whole other view of Disneyland.
It's cool. Very cool. Would you like to see how to do that in your calculator, by the way?
Yeah. But you might. Because if I give you something like this, why don't you calculate 31 factorial?
Huh, that's going to suck because you're like, 31 times 30, 29. That's not that great. We don't want to have to do that every time. So let's see how we do this. I even brought a calculator today.
Yay! Oh, there goes your corneas. Can you see it okay? That's weird. That's better.
Can you shut that mic off for me, please? I just want to... Yeah, why not. Okay, so you have your calculator in front of you. Here's how you calculate a factorial pretty easily.
What you're going to do, you're going to put in the number that you want. So let's say 7. We go over to math. You go scroll over to the PRB.
What's PRB stand for do you think? Probability. Probability.
So it's probably here, right? Probably in there. And sure enough it is.
This is everything we're going to be dealing with in our probability. Probability goes down here. We're talking about probability, as a matter of fact. Go down to the factorial, press enter.
It'll put it after the number that you're putting in there. Press equals, and it doesn't bore you without doing a whole lot of calculation. So that's kind of nice, right?
Instead of having to do 7, 6, 5, 4, 5, 2, 1. This works very well. Did it work for you? Yes.
Okay. So, again, we're going to go math. Probability down to your number four factorial and that will give it to you. If you don't have one of these calculators, if you have one of these calculators, it should still be on there.
I think it is. I hope it's not this one. Yeah, there it is.
Turn it on, but then do you see the... I'll have to move it like this so you can see it. Do you see the factorial button on this?
On this particular calculator, it's above the 3. It's in those yellow letters. On yours, it might be somewhere else. I don't know where. you can come see me at the class and I'll show it to you.
To use those yellow letters, you have to use either a shift or a second function. So to do this one, I press the number 7. I go over to second. Press my second.
Notice how the second is the second. I press the second. I press the second.
Notice how the second is the second. I press the second. I press the second.
I press the second. I press the second. I press the second. I press the second.
I press the second. I press the second. I press the second.
I press the second. Second pops up on the top. And then I press the three. And by pressing that three, it's going to give me actually the factorial above it. It gives me 5,040.
So in this calculator, we're punching in the number, then second, then the factorial, and it'll give it to you either way. Saves you some time, so you don't have to do that, especially with large numbers. However, if we do, man, if we do some large numbers, like 31 factorial.
It's huge. It's a massive number. I mean, that's really, really, really, really big.
Do you know how to read that notation, by the way? 8.22 and then a 33rd? That means that you take that number 8.222838...
8654, you move the decimal places, 33 spots to the right, and that's actually the number. That's a massive number, okay? So your calculator might not actually give you the exact thing, because we're missing some stuff on that. But it's a lot easier to punch in 32 times all that garbage.
Lights, please. Perfect. Okay, no more hairy baby face.
Let's go ahead and talk about these things. So these different arrangements that we're talking about, every different arrangement, that's called a permutation. Can you say permutation for me? Permutation. That was it everybody.
Permutation. Permutation, that's right. Like you're getting a perm and then it mutated. Permutation. That would be awful.
Don't ever do that. Anyway, a permutation is just a different arrangement of items. Permutation is a different arrangement of items.
And for us, we know that n unique items will give us n factorial arrangements. And of course, we mean unique arrangements, right? We don't want to have the same arrangement twice.
That wouldn't be different. The pictures would look the same. However, there are certain cases in which you don't want to find out all of their arrangements. You only need to select a few.
For instance, let's say that you're running for governor. How many counties are in California? Do you know?
how many counties are here i want to say 58 if i'm wrong or whatever say it's 58 okay let's just pretend there's 58 counts i think i'm actually pretty close if there's 58 counties in california and you're going to run for governor are you going to have time to visit 58 of those counties? Probably not. So let's say you're a really lazy governor you're like you know what just pick four out of a hat whatever I don't care. It's gonna be on TV anyway right?
So you're gonna go to just four of these counties in person. So you have 58 to choose from. How many ways could you visit all 58 of those counties?
Are the counties unique? How many ways could you visit all of the counties? You're not going to be able to say like 14 or something like that.
It's not going to work. You have to say 58. How many ways can we visit it? How many choices do you have for your first county?
How about your second one? Are you guys playing along today? 57. And then? Then 55. What am I doing here? 54, 53, 52. What is that called when I multiply every number below a certain number?
Factorial. So the ways you can visit 58 different counties is 58 factorial. Are you with me on that? That's a huge number of ways.
Now, we're not going to visit all 58 counties. We're going to limit it to 4. We're going to go to 4 counties. Thank you. Thank you. There's 58 counties in California, you need to go to 4 of them.
If you're just going to be the governor and pick out counties out of a hat to visit, how many ways, how many orderings can you visit your counties basically? Let's think about it. How many counties are we visiting?
We're going to visit four. And you're going to pick them at random out of a half. Out of a half. So you picked the first one.
How many choices would you have for the first county? You have 58 counties to choose from, right? Are you with me on this?
You have 58 total counties to choose from. How many choices do you have for your second county to visit? Why not 58 again?
You're not going to visit the same county four times in a row, are you? That'd be kind of silly. So, you'd be the dumbest governor in, well, not ever. Never mind, this is political, so whatever. I'm sure there's been dumb governors all the time.
So, you pick the first county, that means that there's only 57 choices for your second county. How many is for the third county? And? What this is saying is that if you have 58 choices for the first one, 57 for the second one, 56 for the third one, 55 for the fourth one, do we have to go on to the 54, 53, 52, 51, 50, all the way down to 1? That would be if you were visiting all 58 counties.
In this case, though, we've got just 58, 57, 56, 55 because we're only going to 4 counties. How many is that, by the way? Can you tell me?
Like this? So if you were governor and you wanted to visit four counties at random, there's 10,182,480 ways you could potentially do that, just visiting four counties. You have 58 for the first choice, 57, 56, and 55 for the last choice. Raise your hand if you understand where those numbers are coming from.
Good, okay. Now, is there an easier way to get here than doing this? The answer is yeah. We actually can do a permutation.
Because I want to be able to use that factorial, I want to apply it to our knowledge over here so we don't have to just think about this all the time and reinvent the wheel every time. Let me tell you some rules about permutations, then I'll show you a formula that's going to allow you to calculate this in one fell swoop. So rules for permutations.
Please make sure you know this is for permutations. First of all... Your items have to be unique.
You can't have any double ups. You can't have any double ups. What that means is with my fruit example at the beginning of class, if I had two apples, if I exchanged those apples, well, that means that the ordering is really not different, right? I'd have two apples either way. The picture would look identical.
Do you follow me on that? So in order for this to really work, to get a different arrangement, you have to have n different items. I'm going to give you another letter.
I'm going to give you the letter R. R is the number of items you're selecting out of N to arrange. So for instance, in our example up here on the board, our N, how much was our N?
How many different counties were there? 58. N was 58. How many did we choose to arrange? Four. R would be four.
Do you see the difference there? So N is the total number of different items. R is the number that you're selecting to arrange.
That's the idea. So R would be the items selected out of N. The number of items to be arranged out of N. The last one.
In permutations, order matters. Now, you might be thinking, well, why are you even saying that? Because all we're talking about is different arrangements, and that's true. We're going to be talking about one more caveat of this probability stuff called combinations in which order is not going to matter.
So. permutations I need you to know that order or arrangement means a different thing. It means that even though you have the same items, arranging them differently makes a difference.
Does that make sense to you? So because maybe on the campaign trail, visiting Fresno before LA would give you a different outcome than visiting LA before Fresno for some reason. Maybe the timing would just work out different. So that arrangement would matter for permutations.
So first, the items are different. Second, R means the number of items that you're arranging out of the N. And third, the arrangement or the order is definitely very important here. Arrangement or order matters. Now we're going to be able to come up with a formula for this thing.
The letter for permutations, it's not very original. It's a capital letter P. So when you want to find a permutation, by the way, does order matter or arrangement matter for permutations? So if you see that, then yes, order matters. arrangements are going to be different.
So this is counting the number of arrangements you have. You're going to have n different items. The n's going to go right here. And you're going to have r items you're arranging.
Now I want to show you something kind of neat with this example here. Watch this for a second. I'm going to bring this over here.
If we visited all 58 counties, revisit this with me, we'd have 58 factorial different arrangements, right? But we're not. We're not visiting that many. This would be 58 times 57 times 56, times 55, times 54, all the way down to 1. All the way down to 1. What we're doing with this formula I'm about to give you is we're saying, okay, you're not visiting all 58 of them.
In fact, how many are you not visiting? If you only pick 4 out of them, how many are you not visiting? 54. You're not visiting 54. You're not visiting 54 different arrangements that you could be doing.
Use your multiplication of fractions and simplification of fractions to look at what's going to happen here. What I'm doing here is I'm taking 58 factorial and dividing away 54 factorial. I had I have 58 factorial different items that I could arrange, but I'm not picking or I'm dividing away 54 factorial arrangements that I'm never gonna go to.
So this 54 factorial, look what happens to this. You know you can cross stuff out, right, fractions? The 55s are gonna cancel, the 53s are gonna cancel, the 52s, I'm sorry, the 54s are canceled, 53s, 52s, 51s, all the way down to the one, that's all gonna be gone.
Do you see what you're left with? We left with exactly what we thought we were going to have over here. Here's how this formula works. The formula says this. I hope you're going to kind of see it on this example once I give you the formula.
The formula says take the number of unique items you have. That's going to give you the number of different arrangements. Stick with me here, folks. If you have n different items, n factorial gives you the arrangements of all n of them, right? That's what we've been doing this whole time.
What you're going to do down here is you're going to take away the ones you don't need. The ones you don't need is everything except the r that you've chosen. These are all the arrangements that you don't want to look at. In our case, in our example, we had 58 counties, sure, that would be 58 factorial different arrangements we could originally go to if we went to all 58. But how many arrangements don't we need? We don't need the 58 minus the 4. That's our 54 of them.
54 arrangements, those are getting canceled out. Those are going to one when we take our numerator and denominator because we're not going to those 54 counters. We're only going to the first one, two, three, four of them. How many of you understand that? Feel okay about it?
Good. So for us, if we wanted to write this out in the example that we just did, you'd have 58. We're finding how many permutations of, not R, sorry. What was our R in our case again?
Four. Four. We have 58 different items. We're finding how many permutations of four items we can make out of that.
So how you'd write this, you'd say, oh, okay, here's 58 factorial divided by n minus r. Notice how that's in parentheses. n minus r, you do that first.
That's order operations, and then you do the factorial. 58 minus 4 factorial. Guess what? Once you do this little step, you've got 58 factorial over 54 factorial, and that's exactly what we did right there.
That's exactly it. This would be the 58 times 57 times 56 times. 55, 54, 53, this is 54 times 53, 52, 51, 50. All those things will cross out. You have multiplication of numerators, multiplication of denominators. You cross all those things out because you have 1 times 1 times 1 times 1. What you're left with is this piece of information, exactly what we had over here.
Nod your head if you're still understanding that. That's kind of a big step for us. So we have, okay. You sure you get it?
You sure? You sure? You sure? You sure?
So here's our formula that we have for permutations. You're right by the way, it's about 50 times. I am?
Yeah. I thought I was. I love being right. Anyway. I am?
Yeah. I thought I was. I love being right. Anyway. Do you understand the formula?
Do you see where it's coming from? What's this? Factorial.
Well, I know that means factorial. It doesn't mean n, right? It doesn't mean that.
But what is this? What does this signify, n factorial? It stands for the number of what's. These are the items, what's n factorial? N is the number of items you have, n factorial is the number of different arrangements you got.
This would be the number of different arrangements if you had n items, all n of them. What's this do for you? This takes away the arrangements that you don't want to consider. That's what this is. That's why you're doing N minus R.
The R is the ones you, listen, R is the ones you want. You want the R arrangements. N minus R would be all the ones you don't care about.
That's those arrangements. That's why they're cancelling out, okay? Would you like to see how to do this on a calculator?
Yes. Of course you would. Lights. My wayla you are my light lady today.
Thank you. Perfect. So we turn our calculator on.
Go up to your main screen. So we already accomplished 7 factorial. We know how to do that.
What you're going to do on this particular type of calculator, you're going to punch in the N that you're trying to deal with. Punch in the N you're trying to deal with. So for us, that would be 58. After that, you're going to go to math because, hey, we're in a math class. You're going to go to probably, you're going to be over here because we're in probability.
We're going to go down to, what's the formula I gave you? What was it? NPR. NPR. NPR.
NPR. It's not here, right? That's just factorial.
We can make this even easier. Go to what you have written on your paper right now, NPR. You've written out the N first. That was a 58. Press that button. It gives you NPR.
Your N's already listed. Then you're going to list out your R. In our case, our R was how much?
It is totally magic. It's not like this thing has programming or anything. I mean, it's just a Harry Potter box, is what this is. Does this have a tube? Perhaps.
Perhaps it does. Do you see down here above your factorial by a couple of spaces you have this NPR? Do you see it? Unless I jiggle it like that? It's worth it.
Earthquakes happen. Okay. So we press our 58, same thing we did in the last calculator.
Press your second button because we're dealing with the yellow letters, NPR. Then you press your 4, same number. Okay, it'll still do it no matter what calculator you have.
So is it possible, can you have much more time? Yes. Is it possible to calculate it this way?
Sure. Yeah, absolutely. This will work every single time for you. The only difference is what if I had like 58P and then I did like 32?
Okay, if I did 58P32, then what you'd be doing is going 58 times 56. times six times 55 times four. That would be very annoying, wouldn't it? Because I'll tell you what, your calculator will not calculate 58 factorial. You can't even do 58 factorial on your calculator and then divide by four factorial. big of a number and won't handle it.
But that way, if you use the NPR, it'll take the shortcut for you, and then you'll be able to calculate your number. I hope you understood what we talked about today. You feel okay about it? Good. The next thing we're going to talk about, we'll do one more example on this next time.
We'll talk about when items are not unique. When items are not unique, and see how that changes what we do. So we were indeed talking about permutations, and what we found out was that permutations are basically just arrangements of unique items. Now, what we're gonna do, I wanna do a little example here.
Let's pretend that for the next election, which we have coming up next year, we're gonna do things a little bit different. ...different in this country, okay? Here's how this is going to work. We're going to take, let's see...
Let's choose ten people at random from our two parties. So let's say that you guys are my Republicans, and you guys are my Democrats, and the back row, you're my independents. You got it?
And here's how we're going to choose a president and a vice president and the secretary of state and what's another one? The what? The general?
Just the general? We're going to say the general, okay? What's your name?
I'm the general. Like the insurance guy. Okay, whatever. So here's what we're going to do.
There's lots more Republicans and Democrats than typically there are independents. I'm going to say you guys pick your four best people. And you guys pick your four best people.
And you guys pick your two best people. And out of those ten people, we're going to make up our leading category of guys. Are you with me on this? So how many people are there total?
Out of that ten, we're going to choose the top four positions out of a hat. So you're going to just put everyone's name in there. And we're going to go, okay, first person I pick out, you get to be president. You like the way that works? I do.
I think that's kind of neat. So first person gets to be president. Second person I pull out of the hat gets to be vice president. Third person I pull out of the hat gets to be treasurer. And the last person gets to be the general.
And the rest of the guys, the rest of the six people, they're better luck next year. Maybe they get to be congressmen or something. Something unimportant.
Whatever. So first person picked out as president, then vice president, then secretary, then the general. Are you with me?
Or treasurer, whatever I said. Do you think it's going to matter what order you pick? you pick people out of.
So if I pick someone out and the first person I pick out gets to be president, are the rest of the people going to care about that? That they weren't picked first? They're probably going to want to be president, right? My question is, if I pick four people out of the hat and put them on the list, and I put everyone's name back, and I pick another four people, if the first four people, let's say we had, this guy's name is A, President A, Vice President C, B, C, D. If those people are going to be president, are the first four people I pick out and I think out a different ordering are these four people going to be a different leading group than these four people in this case this would be vice of the president vice president I say treasurer secretary I remember this guy's the general though so president vice president treasurer general, president, vice president, treasurer, general.
Are these four people different than these four people as far as the rankings go? So in this case, does arrangement matter when I pull out those names out of the hat? That would be our... where arrangement actually makes a difference in what you get, the picture of what you get.
So in our case, the pictures of our leading group of people for our country would be completely different if we were to pull those people out of a hat randomly and assign first, second, third, fourth. That idea of the permutation, that's what we covered yesterday. And what we said was we had this funny symbol NPR.
What was the N? What did the N stand for? Okay, so how many we get to choose from, what's the R?
Alright, so if my example here is there's 10 people who want to be president. How would you want that job? Seriously? I mean, have you seen how much they age over four years? I don't want wrinkles.
My gosh. There are ten people who want to be president, vice president, etc. We need to select four of them to be the president, vice president, treasurer, and the general.
So we want to select four people to lead the country. The first person gets to be the president. The big pres. Second person is going to be the vice president.
Third person is going to be the treasurer. And the fourth person, he gets to be the general. What I want to know is how many ways could we make this happen?
How many different leading groups of four people could we get out of this selection of ten? ...that we have. So the first question is in our case does arrangement matter again? Does arrangement matter?
So the order in which I pick them out is certainly going to matter because the first person gets to be present, second person vice-president, so on. How many people are we choosing from ladies and gentlemen? So that means my N is going to be 10. The reason why I have a P is because the arrangement does matter.
The permutation stands for when arrangement does matter. How many am I choosing out of that ten? One.
Thank you. What we saw last time was that this formula, we really can do it two ways. First way said you take n factorial, you divide by n minus r factorial like this in parentheses and what it said was this. If you really didn't catch on last time, maybe focus up here for a second. that's the number of ways I could arrange my 10 people total, right?
That's everybody. If I was choosing all 10 people, that would be the number of ways I could do it. Remember the n distinct items? These are n distinct people. That's the number of ways I could arrange them.
What is this part? do? The n minus r. Say that again? The ones you don't want?
Yeah. I only want r. So the arrangements I don't want is the n minus r.
If I divide that, that's going to simplify out all those arrangements that I really don't want. It's going to leave me with just my 10 times 9 times 8 times 7 in this particular case. So for us, we could do our 10 factorial over 10 minus 4 factorial, or 6 factorial.
But there was another way. Can you use your calculator to do this? Yeah, why don't you do that right now? I showed you last time how to do that in a calculator.
With the graphing one, you go to the math, because we're in a math class. You go to probability, because it's probably going to be there, and it's on that page. If you have your scientific calculator, you probably have to press a second button.
What number are you going to plug in first? 10. Okay. Then you press your NPR.
And then you press your 4. How much did it give you? 54. You have 54? Yeah. 54? 54?
Yeah. What is that? 5040. Oh, 5040. I thought you said 54. Oh, sorry. There's only 54? 50, oh, that's a lot more.
Okay, that makes more sense. 5040, 5040, what's the 5040 do for us? What's that say? Interpret that for me. What's the 5040?
What's the 5040? 5040. 5040. 5040. 5040. The number of ways of what? So yeah, he's absolutely right.
This is the number of ways that we could get distinct, let's call that the cabinet. I don't know what that's actually, is that called the cabinet? Like the first four people or something?
I know, I think the cabinet's bigger than that, but for right now, in this case, let's call this the cabinet. This would be 5,000, 5,040 different ways you could get these people out of my 10 that you guys selected on your own. So you select four people, you select four people, you select two, you put them in the mix, we draw out four names out of the half, that can happen 5,040 different ways. How many of you understood that?
Feel okay about this? Good deal. All right.
So we would probably get one, right? Andrew? We would probably get one, right? Yeah. All right.
I don't know. Are you Republicans over here? Let's say that I won. I am everything now. Including the general.
Right. The way that this works is that these people have to be different, right? You can't have the same name in the hat twice.
That would kind of ruin the whole idea of different arrangements. Because if you had the same name in the hat twice, you could have the same name in the you could pick out the same name for these two people, right? And if you switch them, it really wouldn't make a difference because the same person would be president and vice president at the same time. Do you understand this concept?
Because I'm going to move on to something just right now that deals with this. Let me say that one more time if you really didn't want to pay attention. If you had the same name in the hat twice, so like Brandon Leonard's in there like 14 times because I really want to be all these things.
Well then, of course, if I got picked twice for president and vice president, even if I were to reverse those names, they're still the same name, right? I still, no matter how you pick that out, and president and vice president, what that does for us is it eliminates some of the arrangements because that arrangement wouldn't actually be different. What I'm trying to say here is that if you have items that are not actually unique, if you have items that are not actually unique, the number of arrangements you get out of that isn't the same as if you had The whole unique item circumstance.
Let me give you one more case here. Did you like the fruit example that we did last time? Yeah.
Yeah. I hope so because I'm about to draw more fruit. Okay.
Let's say that you have, what did I do? Did I do apples? Apples were good. Mm-hmm. A second apple doesn't look so great.
Let's say you have three apples. You got two oranges. And you got five bananas.
Do I have unique distinct items? Are they different? Well, let's pretend this. Let's pretend that these apples look absolutely identical.
I know they don't right now because I'm a horrible drawer. And let's say these oranges in real life, they're identical. You cannot tell the difference between them.
Now, same thing, you can't tell the difference between them. Do I have distinct unique items? Or do I have the same of a couple of them? So they're the same, right? If you can't tell the difference between these apples, here's the point, how many total pieces of fruit do I have?
So if they were all distinct, we covered this last time, if they were all distinct pieces of fruit here, how many arrangements could I make out of this? If I had all different pieces of fruit, how many items do we have? Ten. If they were all different items, all unique items, how many arrangements could I make?
Well, let's do it this way. How many choices would I make? have for the first place?
How many would I have for the second place? Do you see where this is going? How many would I have for the third place?
Do you remember doing this last time? We had ten choices for the first one, nine for the second one, eight for the... Where's it going to end up? How many choices do I have total?
Don't do the math. Tell me a better way to represent that. Ten.
Ten! Yeah, ten. What was the ten?
It wasn't really ten, it was... how do you say it? Ten.
Yeah, ten. Actually, it makes a point. This means factorial.
We have ten factorial, not an exciting ten. But that's how many... be if we had 10 different pieces of fruit.
This is exactly what we covered last time. Were you okay with this? Nod your head if you're still with me on this part. Now, unfortunately, I don't have 10 different pieces of fruit.
What I have is 3 pieces of the same fruit and 2 pieces of the same fruit and 5 pieces of the same fruit. five pieces of this same fruit over here, we need to somehow eliminate those orderings because here's the deal. Let's say I said, okay, now I'm gonna reorder some of these fruits. I want you to close your eyes. So you're gonna pretend to close your eyes, right?
And I do this, and I switch those apples. And you open your eyes. Is it gonna look different to you? If I took a picture of it, would it be a different picture? Is it actually in a different order?
No, no, it's really not. If I switch things that are exactly the same, you're not gonna tell the difference between them, are you? How about these oranges?
If I switch these oranges, are you going to be able to tell the difference in the big picture? Not if they're identical. Or the banana, the same thing. We need to somehow eliminate those arrangements now. Here's how we're going to do it.
Can you tell me how many ways you can arrange three items? How many ways can you arrange three items? Are they the same?
Let's pretend that they're different for now. So I have my three apples. How many ways could you arrange your three apples? Three, four, five.
Sure. How did you get the six? It was just a good guess. Three times four. How many choices do you have for the first one?
Times? That would be three factorial. You guys were right. Those of you who said that, for sure. Now, here's the cool thing about this.
This would be the number of arrangements that actually do not change my big picture. Does that make sense to you? So if I change those apples around...
There are six ways I can do 3 factorial. There are six ways I can do that and not change this picture. For instance, if I labeled this first apple, second apple, third apple, I could do 1, 2, 3. I could do 1, 3, 2. Let me do this, 1, 3, 1, 2, 3, 1, 3, 2, 2, 1, 3, 2, 3, 1, 3, 2, 1, 3, 1, 2. That's six ways I could arrange those.
Is that changing the overall look of it? We need to somehow eliminate those choices and here's how we do it. If we want to eliminate those choices of arrangements that don't really matter, we're going to divide them out. That's how we eliminate things with our factorials. We're going to say, okay, we would normally have 10 factorial.
That would be for unique items. However, I have some things that are the same. One thing that's the same as my apples. There's three of them that are exactly the same.
If there's three of them that are exactly the same, there's three factorial arrangements that don't change my big picture. So I'm going to get rid of those three factorial. Tell me something else I need to get rid of. Why oranges? How many oranges do we have that are the same?
So what else am I going to divide by? Anything else? How many bananas? Come Mr. Dollyman, tell me bananas.
Just kidding. Yeah, you have five bananas here. If you labeled these 1, 2, 3, 4, 5, there would be five factorial arrangements that you could order that would not affect your big picture.
How many people understand this idea? So, every time we do this, we have non-unique items such as our apples, our oranges, our bananas, we're going to divide out by that number of items that are non-unique factorial. If we have n non-unique items, we're dividing by n.
divided by n factorial here and here and here. That limits some of those arrangements that actually don't matter. Let me put this into some in general words here.
For non-distinct or non-unique or non-different items. For n total items, we have n factorial different arrangements we could make. Every time that we have some set of items that is not distinct, we're going to divide that out.
So n1 factorial, n2 factorial, blah, blah, blah, all the way down to where we stop, where n1, n2, n3 are the count of non-distinct items, or items that are the same. So n total items, n1, n2, n3, and so on, are numbers of non-distinct items. Okay, so let's recap a second, go over everything we were just talking about. First thing, different arrangements are called permutations.
We're trying to look for different arrangements when we're talking about that. If we have a distinct set of items, such as here are people, the number of items... that we have to choose from gives us n factorial different arrangements, for sure. If we're only looking for a certain number of those people, we're taking nPr or selecting R out of n total choices, and this is the way we show we don't want the rest of those arrangements. We can do this on a calculator.
calculator, we can do that with a formula, that's great. If we happen to have some non-distinct items, so items that are the same, the way we show the different arrangements, or the ones that really matter, is we have our n factorial, that would be everything if it were all distinct. But we're going to take away or eliminate some of those arrangements here, here, here, of those items that are the same, because they don't really change the big picture of my arrangements. Would you raise your hand if you're still with me on this idea? Good deal.
Let's practice one of these to really get the hang of it. Oftentimes you'll see this type of problem with the situation here. It's a question that asks, how many words could you make up if I just gave you a set of letters? So what's your favorite word? Statistics, of course.
I know I can read your mind. So our favorite word is statistics. That's my favorite word. Favorite word is statistics. What I want to know is how many ways could we arrange the letters to get different quote-unquote words.
Now, some of them aren't going to make sense at all, but how many ways could we arrange the letters to get different words? How many different ways could you arrange these letters? Let's look what happened over here with this problem. Let's actually calculate this thing, then we'll move back over here.
What does this do when we're dividing out the 3 factorial, 2 factorial, 5 factorial? By the way, what does the 10 factorial actually mean? Don't all talk at once.
It's really annoying, you know. What's the 10 factorial mean? Say what now? Minus factorial, 8 factorial, 10 down.
So it's a multiplication problem, right? So the 10 factorial means where do you start at? 10. And you multiply by what number next? 1. So this means, that means that.
That's what, we write it like this because it's a lot easier, but that means that right there. You okay with that so far, right? What does the 3 factorial mean? So this little piece gives us 3 times 2 times 1. That's just that piece.
We're multiplying all that stuff. What's the 2 factorial mean? How about the 5 factorial?
Well, we know that means 5 times 4 times 3 times 2 times 1. So we had our 3 factorial, our 2 factorial, and our 5 factorial. Do you see what's going to happen when we divide out these numbers of arrangements? Do you see that our numbers are going to ultimately get a lot smaller?
Because we're dividing, right? Can you see some things that simplify? Sure, what simplifies? Oh, this is all great.
Phi factorial is right here, and phi factorial is right there. Those are completely gone. Anything else? Remember how to simplify some fractions? Look at the factors on the numerator and denominator that are the same.
What else do you have up here? Say it louder. I can't hear you. I can't hear you. Say it.
What do you simplify? Those are fractions. Come on.
3 and 9. 3 and 9. Okay, that's great. How many times does 3 go into 3? 1. And into 9?
So what do we have left? we need to multiply. 10 times what?
3 times 4 times? Times? Anything else?
We simplified everything else. Have you done that already? Let's do 10 times 3 times 4 times 7 times 3. Clouded?
If you did 10 factorial, you're going to get a huge number. Huge number. But when we consider that we don't have distinct items like apples, oranges, bananas, that really limits what we have. There's only 2,520 ways we can arrange these pieces of fruit that would be different. to see what happens with our statistics problem.
Do we have the same scenario we have had over here? We have a certain number of items. How many items do we have in this case?
Items are letters here. How many letters do we have? OK.
Wow, that's weird. Do we have any of the letters that are the same? So for instance, if I switch this T and this T, if I just went, would you be able to tell the difference in this word?
Those T's are the same, same thing like our apples and our oranges. So we would initially have 10 factorial different items to arrange. If these were different items, that's how many arrangements we would get out of it.
But now we've gotta divide by all the items that are the same. Give me one of the items that are the same. S.
How many S's? Three. What are we going to divide by? Three. Just three?
Three times one. OK, three factorial. Anything else?
T's. How many T's? Three. That's another three factorial. Anything else?
Three. That's another three factorial. Anything else?
Three. i's i's? 2 2 How about anything else? How about the a?
a only happens once. If you divide by one factorial or divide it by one, that really doesn't do anything. So we're just counting it up.
We're counting it up greater than one. Well, any repeats that we actually have. Why don't you see if you can figure this one out?
Go ahead and write it out and see how many arrangements we get that are different out of the word statistics. If you finish that quickly, why don't you try one more? How many ways could you... well you could have a lot of kids, you could have both kids.
How many ways could you plan to have a family with seven girls and five boys? Try that one if you finish your statistics first. I'll be walking around.
If you need help with this, let me know. Good, lots of good work so far. So we do have to write out the 10 times 9 times 8. We have to write out the factorial because we're now simplifying our fractions. So on our fractions we have our 3 factorial, which in this case becomes 3 times 2 times 1. We have another 3 factorial. And then we have a 2 factorial.
Now the 1's don't do a whole lot for us, but the 3, the 2, the 3, the 2, and the 2, those should all simplify. If you simplify these the way I'm looking at it, I'd say these 2's are gone. We're going to simplify those out.
Maybe this 2 and this 4. Maybe this 3 with the 9. The reason why I'm doing that is because I like to also match up 3 and 3 and 2 and 2. So these 3's can also be eliminated. Remember we're making 1's out of this stuff. And this 2 and this 2 can be eliminated. We're making 1's out of that stuff.
Then I just have to multiply 10 times 8 times 7 times 6 times 5 times 3. And I can get the appropriate answer. Did you guys all do something similar to that? What did you get? When you multiplied. Did you multiply?.
Okay. Five zero.. Oh, you all got it. 50,400 different words you can make up from statistics. Fortunately, we chose the best one at the beginning, just statistics.
That's great. If you had had 10 factorial different words, that would be way bigger than 50,400.. Okay. So, we're going to do a little bit of math. We're going to do a little bit of math.
We're going to do a little bit of math. We're going to do a little bit of math. We're going to do a little bit of math. We're going to do a little bit of math.
We're going to do a little bit of math. We're going to do a little bit of math. We're going to do a little bit of math. We're going to do a little bit of math.
We're going to do a little bit of math. We're going to do a little bit of math. How about if you start on this one? How many ways could you plan to have a family with seven girls and five boys?
How many people are we having in this family? How many kids? So if you had… 12 distinct items, you would get 12 factorial from them.
So that would be 12 ways, if you would, like I said, how many ways could you arrange 12 people's names? That would be it. But for our case here, we're considering our girls and our boys to be the same.
So we have non-distinct items. What do we need to divide by to get rid of some of these choices? I'm going to leave it to you to figure out what that one is, simplify that on your own.
What I need to know is how many people feel pretty good about this idea of eliminating these non-distinct choices. Guys on the left-hand side, are you okay with this? Okay, good. Is there ever a case where order doesn't matter? Like arrangements?
What if you said, like a city council. On a city council, he says like... Is someone actually in charge? I actually don't know, so you should answer that for me. Is someone actually in charge of a city council?
There are? Oh darn it, okay. How about like a...
let's say you're just going to make a committee. We have lots of committees on this school. Maybe you've been on a committee.
Is anyone in to decide, is like the leader person, is there an ordering of committee members? If everyone's just on a committee, the answer is no, I know that one crap. Like everyone's on the same level, all your input is equal.
No one's the president, vice president. There's no hierarchy on a committee. So here's the question.
If you're going to pick out a committee, like a committee of people, I'm just going to pick your names out of the hat. If I just say there's ten people who want to be on a committee, we're going to pick out eight people. So only two people are excluded.
If we're going to pick out those eight people just by pulling out names out of a hat, does it really matter whose name is first? Does it matter whose name is second? If you're all just going to be on... The committee, it's really not making a difference.
So we need to kind of determine what we do if ordering doesn't make a difference for us. So sometimes order doesn't matter. You know what, that one didn't seem to make a whole lot of sense to you, the whole committee.
Let me give you another instance, okay? Let's go down to Magic Mountain for a while. How many rides do they have in Magic Mountain? Let's say like 52. Like 52 rides, one for every week of the year. So, you go there every week for a different ride.
So they have 52 rides. And you say, I don't really care the order that I'm going on those rides. All I care about is...
I want to go on five rides. Just five. I don't care which one comes first.
I don't care which one comes second. I just want to ride my five rides. Because at the end of the day, you just want to say I rode the Riddler and the Tatsu and whatever three other. I only went on there like twice. The what now?
Superman. X2. Batman. X2 and Batman. Okay, those are the rides.
That would be five rides. Would it matter the order you went on? I don't care as long as I go on the rides.
So in our case, the arrangement of those rides going on Superman first versus going on Batman first really won't make a difference to us. We just want to go on five rides. You get the idea. So the arrangement really doesn't matter. So let's say that you want to...
Go on five rides out of, I don't really think there's 52, let's say there's like 25, out of 25 rides at Magic Mountain. You don't care about the order on which you go on these routes. So the question is, how many ways could we ride five rides?
Or how many combinations of five rides could we select? How many combinations of five rhymes could we select? Combination means I really don't care the order.
What I care about is that we just pick the five rhymes. There are some rules for combinations. First rule, first thing you've got to have, again, you have to have n different items because we found out if we have non-distinct items, things change up a bit.
So with combinations... Again, we're back to the n different items. So our rides, we're talking about five different rides here.
We're talking about 25 total different rides. Are you with me on this? You're not listing rides twice over here. You have different rides. So n different or distinct rides.
Just like permutations, we're going to be selecting a certain number of them to write or to have in our combination. So we're still selecting, what was that letter we used to select a certain number? R. R of them. And third, for a combination, the arrangement does not matter.
So, Ryan, ABC, going on A first, then B first, then C first, would be the same as CAB, or any other ordering of those. What the combination says is I'm just trying to pick out five rides. I don't care the order. As long as I'm going on ride A, B, and C, or I'm going on ride C, A, and B, those are the same for me. I'm going on those three rides in total, right?
At the end of the day, I went on the same exact three rides. Are you with me on this? In our case, we have five rides. Now let me give you the formula for our combinations. I'll show you how to use it on a calculator and we'll be done for our day.
So with our combinations, it looks really similar to the permutation. We still have an N. We have a C now instead of that P because instead of a permutation where arrangement matters, we have a combination where arrangement doesn't matter.
But we're still picking out our items. On the numerator fraction, we still have our... N factorial. That would be the total number of arrangements with N distinct items, and that's what we have. However, we are going to have to eliminate a lot of stuff on the bottom.
Now, when we do this... n minus r factorial. What they got rid of was all the arrangements that we didn't want that did not include our items. That's the same as a permutation.
Right now you have a permutation. Now, if you're still with me, that's a permutation. How do we make it a combination?
As we take out R factorial, here's what this says. This is kind of combining a couple of ideas here. This is treating those R items as non-distinct. It's dividing out, treating them like non-distinct because look what happens. This says if CAB is the same as ABC, that's treating those three items as non-distinct items and that's how you get rid of them.
Do you see the similarity there between them? That's what that's doing for you. Now can you do it on a calculator?
Can you do it by hand? Sure. But it's a lot easier on your calculator.
We're going to show this real quick on your calculators here. Can you see me? Oh wow, that's really good.
Where do you go to first, ladies and gentlemen? Math and it's going to be where? Our probability.
Yeah, you might have already seen it actually. We go to the probability and what we're looking for on this screen is, here's for factorial if you want to do it by hand, you can do that. This one was, did arrangement matter for this one or not? This one, the NPR. Does arrangement matter for that?
Yes. Does arrangement matter for this one? No. This one's called a, what's that called again?
A combination. And this is a? A combination.
Good. So we're picking out, I'm going to go back to my original screen here. We're picking out how many rides did we want to choose from?
Five out of a possible... 25. So what's going to come first, the five or the 25? 25. We have 25 rides we can choose from. We're going to find out how many ways we can ride five rides out of that.
We go down to... This would be if the order mattered, the order of our rides mattered. That would be it. If we say, I don't really care.
I really just care about going on five rides. We're going to pick the NCR. And then what do we put after that? 53,130 ways we could go on five different rides out of 25. If you want your handheld calculator, the scientific one, it's the same basic idea. We do our 25. On this case, you had to press the second button to get down to your NCR, which is above the number 8. And then we press our 5, but we'll get the same exact answer.
I'm going to feel pretty good about what we talked about today. Do you understand the difference between a permutation and a combination? Do you understand how to use your calculator to find both those things? Good.
If you don't, if you're having trouble with that a little bit later, come and see me in my office or my math lab hour. I can show you how to do that.