Hi! So we're starting counting methods today. We're doing topic one, introduction to the fundamental counting principle. So introduction to the fundamental counting principle, which sometimes for the sake of not writing it out, I will write the FCP. And our goal today in this lesson is to determine the fundamental counting principle and to use it to solve problems.
problems. Some of it is intuitive and some of it's a neat strategy. So let's start.
We're going to explore how many different outfits are possible. So we have American Eagle Outfitters is a popular clothing store that carries both men's and women's fashions. Suppose a customer purchases the five color coordinated items shown below.
Three pairs of shorts, navy, black, and gray, and two t-shirts, white and red. list the possible outfits consisting of a pair of shorts and a t-shirt so we'll start with navy we have navy shorts with a white shirt then navy shorts with a red shirt then we'll have black shorts with a white shirt and black shorts with a red shirt and then we'll have a gray shorts with a white shirt and gray shorts with a red shirt so those are all the different outfits now we can tell how many by counting one two three four five six so there'd be six outfits okay we can also organize this information in a tree diagram whoops this is really touch the wrong thing sorry about that guys so we can organize it using a tree diagram and this can be used to organize all our possible outcomes. So we're going to fill in the blank of the tree diagram shown below. So for shorts we had navy, black and gray. For t-shirts we had white and red, white and red, white and red.
And this allows us to build out all of our combinations. So navy and red, black and white, black and red. and red gray and white gray and red okay and as you can see again we have six outfits now for this one it's pretty straightforward a tree diagram.
Sometimes a tree diagram, particularly when we combine later counting methods and probability, a tree diagram can be very very useful. But it also can help you map out things when you have more than one options. And here we just have two things, shorts and t-shirts, but sometimes you'd have shorts, t-shirts, and shoes. So the T diagram can help us map them all out. Explain how you could numerically calculated the number of outfits without actually listing the outfits or representing them as a tree diagram.
So mathematically or numerically, what we have is three pairs of shorts times two different t-shirts. And if we multiply them, we will get six. And that will get us our number of outfits. Now, the customer has a choice of purchasing another color-coordinated outfit. a t-shirt or a pair of shorts.
Make a prediction about which purchase will give him or her the greatest number of outfits. So it can either buy another pair of shorts or another t-shirt. It wants to know which one is going to give them the most outfits. Well, let's see. If they purchase another pair of shorts, they'll have four pairs of shorts times two t-shirts, which is equal to eight, right?
So if he buys shorts, he'll have eight options. If, however, he buys the shirt, he's going to have three pairs of shorts times three t-shirts, which will give him nine outfits. So this is the shirt.
This is by increasing the shirts, right, to three. Here we increase the shorts to four. So this one will give more outfits. More outfits. And you could do that just by multiplying without having to draw all those tree diagrams to help you figure it out.
Okay? So this becomes a strategy. Okay. So now we're going to talk about counting methods.
Introduction. A sample space is an organized listing of all possible outcomes from an experiment. Two common ways to organize a sample space are to use an outcome table or tree diagram. So we saw the tree diagram in this example. And we'll do some stuff with outcome tables as well.
They're particularly useful when we're doing like rolls of dice and cards. and things like that. In an outcome table, it's a table that lists the sample space in a very organized way.
In tree diagrams, each branch represents a different outcome of the sample space. If only the number of outcomes as needed, we can also use the fundamental counting principle, FCP. Now the fundamental counting principle says that if there are A selections of one item, B selections of a second item, and C selections of a third item, then the number of arrangements of all three items is A times B times C. So we're just basically multiplying. And this can be...
for any number of items okay so very neat strategy okay for example let's do some examples using outcome tables and tree dice diagrams to represent sample space. So the menu in an oriental restaurant lists two meat dishes, two vegetable dishes and three rice dishes. How many meals are possible consisting of a meat, vegetable and rice dish?
So let's do a tree diagram. So we have meat, veg, And rice. You're going to want to space your first one out so that you have space for the other criteria.
So we have meat one and meat two. Okay. Now in the tree diagram, that then extends and we have two vegetable choices. So let's just highlight our two meat dishes, two vegetable choices, and three rice dishes. So in the veg, we have V1.
And V2, V1, and V2. And then we have three rice dishes. So we have 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3. So let's list those out. So we have rice 1, rice 2, rice 3, R1, R2, R3, R1, R2, R3.
R1, R2, and R3. Now if we look there, this tells us all of the sample spaces and all the possible variation. We could have meat 1, veg 2, and rice 2. We could have meat 2, veg 1, and rice 3, depending on how we figure that out. Use the fundamental counting principle now to determine how many meals are possible. So we have two for meat.
times 2 for veg, times 3 for rice. And we multiply that, and we get 12. And if we look up here at our final column, and we count our number of final roots, we get 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. So it matches up with the 12 possibilities that are listed here. Okay, so either way got you to the answer. Now, that's fine for this. Sometimes we don't want to do a tree diagram.
It's going to take us too much work because there's so many options. And using the fundamental... counting principle to solve a counting problem that has very large numbers as its answer is much more preferred.
Okay, it works on a small scale, it works on a large scale. Excellent. So let's look at this situation. situation here.
We have a combination lock with 40 numbers. To open the lock, you rotate clockwise to the first number, rotate counterclockwise past the first number to the second, and then clockwise back to the third. Most combination locks work that way.
Question A says, how many lock combinations are possible if, and this is important, if any number may be repeated? So we can use all 40 numbers each time you can have a common Nation of 30, 30, 30. Okay? So we have to think about each choice. In the first possibility, we have 40 options.
In the second possibility, we have 40 options. And then in the third possibility, we have 40 options. I just want to take that away because I want to make sure we don't add numbers we don't need. Which is the same as having 40 to the power of 3. If we multiply those together, 40 times 40 times 40, and you can try that out, you get 64,000.
So that's how many combinations. If you can repeat the number. If it takes 5 seconds to dial a complete combination.
How long would it take to dial all the possible combinations? Well, then you would take 64,000 times 5 seconds. And you're going to get 3,200,000.
Or 320,000. seconds. Now I can convert that to minutes by dividing by 60. So I get 5,333.3 minutes.
And I can convert that to hours by dividing by 60. And I would get 88.8 hours. This is just to give you more scope so you understand what the time is to do that, to put in all those combinations. And if I divide hours by 24, I would get 3.7. days. So if you forgot your combination and you wanted to try all of the combinations of a combination lock, and of course this is saying you're not sleeping or stopping to eat or go to the bathroom, it would take you 3.7 days.
Okay, so that's a bit of time. Now let's do another fundamental, let's do another fundamental counting principle problem. Of course I would have to flip it to the right page. How many combinations... Combinations are possible if the middle digit is 12. So what that tells us is the first option is 40, right?
Now the second digit, we only can have 12. There's only one option, so it would be times 1 for that. And then our third option... again, we have 40 choices.
The only limit was on the second digit. So if the second digit, say you forgot your combination, you're saying, well, I got to figure it out. So I remember the second digit is 12. How many would I have to try out?
Well, 40 times 40 gives us 1,600 options. Okay? So when you're dealing with the fundamental counting principle, you address...
the restrictions first. So the restriction in this case is that 12, okay? The fact that there's only one option for the middle term, okay? So make sure you take that into consideration. Now let's look at some other options there.
How many combinations are possible if the first number is 28 and the last is 47? Well, that means for the first option, 28, there's only one option. Okay? The second, we could have 40 options because we have no limits on that one. And the final option, 47, the third option, again, only has one option.
So that would give you 40. So say you remembered your first and last number. Well, 40, you could probably try all 40 different ones in the middle and figure it out. Okay?
I actually have a combination lock like that because my son wrote the number down weird and I can't remember the last one. can't read the middle number so I could probably travel through all the possible combinations and try and figure out and it's actually the middle number that I don't know am I going to probably not because we don't need a lot this year okay all right how many combinations are possible if the numbers are 12 28 and 47 but not necessarily in that order so I know that I'm either going to have 12 times 28 times 47 or I could have 28 times 12 times 47 or I could have 47 times 28 times 12. I could have a variety of choices right now I can do a tree diagram. So just listing out my possibilities, right? So I could have, I know I'm going to have either 28, 12, 47. And then my second digit is going to be one of the ones that's. not.
So from 28, it could be either 12 or 47. From 12, it could be either 28 or 47. And from 47, it could be 28 or 12. And then I have my options for my third, right? And it's actually, for my third, I only have one choice. I don't have two choices, right?
Because there's only one more option. This would be 47. This would be 12. This would be 47. This would be 28. This would be 12. And this would be 28. So looking at that, my only options, the total possibilities would be 1, 2, 3, 4, 5, 6. so I could do tree diagram and get six options. However, I can also use my fundamental counting principle. In the first option, I have three, so using fundamental counting principle, three choices, but now I've used one. So my second choice, I only have two options left.
And then for my third, I would only have one. When I multiply those out, I get six. Because as you can see, your options reduce when there are limitations and restrictions. Okay? Alrighty.
I like these things because I'm a big puzzle freak, so understanding some of these principles helps you solve puzzles better. We're going to do some more examples, and we're going to look at some restrictions and see what happens. Okay, example number three. Using the fundamental counting principle to solve a counting problem.
So, a country's postal code, we all write letters, right? Mm-hmm. consists of six characters. The characters in the odd positions are uppercase letters, and the characters in the even positions are digits.
So my postal code is T5X5N7, so it goes letter, number, letter, number, letter, number. Okay? Now, in the even positions, the odd positions are letters, uppercase letters, and the even ones are digits, zero to nine. All right.
From that, how many postal codes are possible in this country? Now, it doesn't have any restrictions on repetition or letters you can't use. So we're going to assume that we can use anything. So in the first position, well, it can be either A to Z.
So there are 26 options. And in the second position, it can be 0 to 9. So there are 10 options. Third position again, A to Z.
to Z, 26. Fourth position, 0 to 9, so 10. Fifth position, A to Z, so 26 again. And then finally, the last digit, 0 to 9, which will be 10. When we multiply those, we get 17,576,000. There's lots of possible postal codes.
Okay, that's just for an example of a country, a fictitious country. Canadian postal codes are similar, except we do not use the letters D, F, I, O, and U, because they kind of look like either E, they could be mistaken for E or V. The U and the V could look like the same letter, or the number 0 or 1. So they just took a few out. They didn't need 17 million poster codes.
So they said, you know what, we can take these letters out because it might be confusing, particularly when we're dealing with people handwriting. So if we look at that, they removed five letters. So that we don't have as much confusion. So in Canada, when we're looking at these six positions, that means we only have 21 letters that we're using. we still have 10 digits, 21 letters, 10 digits, 21 letters, and 10 digits.
And that gets us a resulting 9,261,000. choices. We're still good for lots of post-it codes. Okay. Alrighty, so how many less postal codes are there in Canada compared to the country in Part A?
Well, 17,576,000,000 minus 9,261,000. So we have quite a few less, 8,315,000. thousand less. Okay, so that's our introduction to the fundamental counting principles. A quick little lesson to kick us off.
We're going to do probably a second lesson today that's a little shorter too. So the fundamental counting principle says that there are A selections of one item, B selections of another item. And C selections of a third item, we just basically can multiply by A times B times C.
So you just basically have to, the hard part is figuring out how many choices in each step. The fundamental counting principle applies when... Tasks are related by the word and.
Okay? When using the fundamental counting principle, address the restrictions first. So think about the restrictions before you start multiplying.
Otherwise, you will not. not get the correct answer. We can use lists, outcome tables, tree diagrams to help us solve things.
You can use them in combination, particularly convenient when we want to list all the possible outcomes. Okay, so that is our first lesson.