we are going to cover objective 34 we are going to solve problems involving permutations and combinations okay um before we begin let's do a little bit of an activity okay I need five volunteers okay one two three four five okay sorry I didn't make it all the way over here all right we can do six we'll do six um six people in the back right there in the corner all right so I would like for the class I want to take a picture of these six people in a line so you can't like put two in the front and two in the back and you know blah blah blah I want them in a line and I want to know how many different orders can I put them in to take a picture of them so I'm GNA hand it over to you as the class and I want you to figure it out start start rearranging them how many different ways so you're in an order right now all right so let's go to a different order all right so there's two ways that would be a third way right Fourth Way fifth way sixth way seventh eighth way ninth way oh wow we're really switching it up 10th Way 11th way 12 13 14 so my question is is how many different ways do you think that I could rearrange them you think 36 and you think more than 36 okay so Emily says more than 36 all right thank you very much you can sit down so how many think 36 ways okay how many think it's more than 36 because that's what Emily says anybody have any other guesses 42 720 you think it's you think it's that many you think so that's because you've been competing on math team okay um all right so let's talk about this and let's see okay let me give you a couple definitions first okay let's talk about the difference between a permutation and a combination a permutation is an arrangement of a a group of objects or people in a certain order and a combination is an arrangement or selection of objects where the order is not important so I'm going to tell you the similarities between permutation and combination is we are both going to be talking about ways you can arrange or ways you can choose but the difference is and it's going to affect your answer is whether or not you're putting them in a specific order or not if you're placing them in a specific position or order then that would be considered a permutation if you are not putting them in any order if I'm just choosing the people or selecting the objects then that would be a combination so my question to you is our example back here would that be considered a permutation or a combination it actually is a permutation because you know what I was doing with the six people I was placing them in positions in order does that make sense like I took a picture of them in one order and then I actually took them I didn't take them you took them and rearranged them in a another position or order do you understand permutation is an arrangement of a group in a certain order so I was placing them in a certain position to take the picture right when someone says let's line up to take a picture what does everybody do okay you stand here and you stand here and let's pull our other people here and our short people get in the front you're placing them in a specific position right in order to take the picture that would be considered a permutation now what if I had the six people and I said I just want to choose two out of the six I just want to pick two out of the six that would be considered a combination because I'm not putting those two people I'm just choosing to right I'm not putting them in a specific order I'm just picking two from the six that's the difference and that's where the mathematical side of this is going to be easy the formula is easy the way to put it in the calculator is easy the difficult part of it is choosing because sometimes it gets a little complicated depending on your opinion of whether it's permutation or combination okay um so what you actually did let's look at factorial first Yep this is a definition of a factorial and it is denoted with the exclamation point it's the product of all the counting numbers beginning with in and counting backwards to one okay it's the product of all the counting numbers what are counting numbers one two three four five six so basically whole numbers except for zero right and you're going to start with that number and you're going to multiply because it says product and you're going to count backwards all the way to one so if I looked at five factorial what would that equal oops what would that mean if I'm going to start with five and I'm going to count backwards so five * 4 because it's a product we're multiplying time 3 * 2 * 1 and so five factorial is going to be equal to 120 okay you actually did six factorial back there okay in order to figure out the number of ways we can take those six people and rearrange them in that order is six factorial 6 * 5 * 4 * 3 * 2 * 1 which is equal to what oh I believe Ryan deserves a piece of candy for that okay I might give Emily a piece just for everybody's like 36 yeah 36 don't jump on on the boat of 36 I appreciate you saying uh I think it might be a little more than 36 okay there's actually 720 different ways we could rearrange those six people in that order does that make sense and we did that by doing six factorial you're welcome you know where it's at all right so Sky wants to show us let's look in our calculator Sky says there's a way we can do this in our calculator so Sky I would like for you to explain let's start with five factorial okay what would I do to get five factorial in our calculators so you're going to hit five and then you're gonna go to math what is what is prb stand for probability okay and it's the fourth one it's the exclamation point and so five factorial when you press enter you get 120 okay tried doing six factorials six math and then Arrow over to probability and choose the fourth option Sky there you go any questions about factorials all right so we're goingon to actually look at some um formulas for permutations and it looks like this um I do believe this is on your formula sheet yes I do agree no P that is permutation P stands for permutation when n is the number of total possibil ities like how many objects or how many people are you working with and R stands for the rate at which they are chosen okay n is the number of total possibilities and R is the rate at which they are chosen so in this case we did say the people in the back was a permutation right because I was arranging them in a specific order so my question is is what would n be in this case how many people are we working with we were working with six people and the rate at which you're choosing them like if I have six people and I'm choosing two of the six then my rate is two because I'm choosing two but my question is is when I'm rearranging all six what's my rate I'm choosing all six right I'm rearranging all six so it's going to be a permutation where I'm working with six people and I'm rearranging all six people in a specific position so if we worked this out in our formula it would be six factorial divided by 6 - 6 factorial and everybody says well wouldn't that mean you would be dividing by zero and we know it's all fun in games until you try to divide by zero and that just can't happen but zero factorial there's no such thing really as a zero factorial um because when you do a factorial you only multiply down to to one right you never multiply by zero so really and if you put it in somebody try to put in 0 factorial and tell me what it's equal to it's equal to one so really it's 6 factorial divided one and anything divided by one is just six factorial and so doesn't that make sense that when we rearranged and I wanted to know the different possibilities it would just be six factorial which is equal to how much 7 20 ways okay any questions on that so let's look look at example two let's look at example two there are 10 finalist in a figure skating competition and I want to know how many ways can gold silver and bronze medals be awarded so my question is is would this be a permutation or a combination why permutation right and I'm taking those 10 people I'm only choosing three right but I'm placing those three in a specific position or order right I'm giving one the gold medal I'm giving one the silver medal and I'm giving one the bronze medal okay so in this case what would n be n would be 10 and your rate would be I'm choosing how many three to give the top three Awards to okay so if we did our formula in factorial over nus R factorial this would be 10 factorial over 10 minus 3 factorial now I want you to understand before you put this in your calculator what this means what is 10 factorial mean 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * one what does seven factorial meantimes five Time 4 * 3 * 2 * 1 can you tell me something about this 7 over seven is equal to one cancel cancel cancel cancel cancel cancel everything from seven and down is going to cancel so really what's it going to be equal to 10 time 9 * 8 which is how much 720 now that wasn't just a quinky dink that it was 720 on six factorial and then 720 for this guy right here um but there are 720 different ways that we can take 10 people and award them gold silver and bronze okay sometimes with permutations and this only works with permutations I like to do blanks for example I know that I'm going to have three blanks here why three blanks I'm going to have a gold medal a silver medal and a bronze medal how many possibilities and this is based on the fundamental counting P um principle and I think you probably covered that in maybe seventh grade which I know was a really long time ago but that's okay okay how many different people could I put in the first position 10 well okay let's just say we award Gabe the gold medal yes okay how many different people could I put in the second position nine and then how many different people could I put in the bronze eight and then that would give you 720 as well does that make sense so sometimes you will see that as well we we'll be using the blanks um along with the form formula um in an activity that we do a little bit later on today all right so let's go to our combination formula okay still n is going to be the same thing it's the total number of possibilities that you're working with and R is the rate at which they are chosen okay the formula says n factorial / nus R factorial time R factorial so it is slightly different with a combination okay everybody got this down so let's look at an example I'll let you kind of just digest it and you can pair for you don't have to write word for word so a group of seven students working on a project needs to choose two students to present the group's project how many different ways can we choose the two students do you agree that this would be a combination why comination the order doesn't matter and um you know you're just choosing two students does that make sense we're not placing them in a specific position we're just choosing two out of the seven okay so if we looked at the formula for this what would n be seven and the rate at which we're choosing we're choosing two out of the seven okay so this would be n factorial divid n minus r and before you put this in your calculator let's just look at it by hand first because I had some people earlier that weren't quite getting it in the calculator so I like for you to understand how to do it by hand so not everything some of it will cancel um but let's look at it so this will be 7 factorial divided by 7 minus 2 factorial times 2 factorial so really this would be 7 factorial / 5 factorial time 2 factorial so let's talk about what seven factorial is 7 * 6 * 5 * 4 * 3 * 2 * 1 how about what's five factorial 5 * 4 * 3 * 2 * 1 and then we're also going to Times by two factorial which would be 2 * 1 okay so tell me what will cancel 5 4 3 2 1 so what will I have left 7 * 6 divid so that's really 42 ID two which would be 21 okay let's put it in your calculators and see if you can do and I wouldn't somebody tried to do this part in their calculator try it and see what you do I'd rather put this one in my calculator we had some issues earlier what would you have to do can you tell them where would the parentheses okay around the numerator and the denominator but especially the why because there's more than one okay so you got to definitely have an extra set of parentheses even if you do this one right here you got to have an extra set of parentheses around the denominator and it wouldn't hurt for you to have it around the numerator as well did everybody is everybody able to get it in their calculator and come out with 21 good okay so let's see okay let me pause and let me come see your calculators here we go let's look at just a few more examples so there's five total on today's lesson so we're almost finished with the lesson for today okay how many ways can you arrange seven Shoppers in a line at a checkout counter how many ways can you arrange seven Shoppers in a line at a checkout counter permutation combination why permutation well but I could what if if I wanted to um what if I had seven Runners and I want to pick the top three that would still be permutation okay um it's the fact the difference between permutation and combination is are you placing them in a specific position or order so when I have those seven Shoppers in the line at the checkout counter do I have to place them this person you check out first this person you check out second am I putting them in a position yes you are you're putting them otherwise somebody's got to go first because does that make sense when you check it out otherwise there would be a brawl in the grocery store okay I mean it'd be like everybody like no I'm first okay you got to put somebody first somebody second somebody third somebody fourth somebody fifth somebody sixth somebody seventh okay but in this so it is a permutation because the order is important for permutation do you see how I remember that that order is important so important the p and permutation does that make sense the order is important for a permutation okay the P helps me remember that so my question for this if we use the formula what's my um total number I got seven people what's my rate seven I'm using all seven and I'm putting all seven in that order so if I did this this would be n factorial Over N minus r factorial so n factorial is 7 factorial over 7 - 7 and what is zero factorial so this is really just s factorial divided one which is just seven factorial okay and so what would it be 5,000 sorry 5,40 ways okay any questions on why it's a permutation and why we're doing seven with a rate of seven at a time because I'm choosing all seven question are you good okay so this is where what did I tell you today the tricky the math is not the hard part okay It's tricky to know whether and we're going to practice whether it's important or not important so here's what I'm going to say if you can take the objects and specifically put them in a specific position or assign them a spot does that make sense like if I'm saying okay I got this pencil and this pencil and I'm going to place this one here here and this one here and I'm going to put them in a specific order pencil pencil calculator does that make sense then that would be considered a permutation if it's a combination I'm just choosing or selecting I'm not placing them anywhere okay I'm not placing them anywhere I don't know which one's which all right so let's just look at more examples and maybe this will help okay how many ways can you select four of 13 different colored balloons how many ways can you select four of 13 different colored balloons so Nate I'm going to choose four of the balloons am I placing the balloons in a position or am I just choosing them so would it be a permutation or a combination It's a combination because I'm just choosing them okay and it doesn't really matter what which ones I pick okay I'm just selecting them okay so this would be considered a combination what goes first and this is going to be super important in about three seconds 13 your total number goes first and your rate goes second and if you don't do it in that order it will be wrong okay so n factorial / nus R factorial * R factorial so n factorial would be 13 factorial divided by 13 minus 4 is going to be 9 factorial times 4 factorial any questions on that now you can put that in your calculator but let's think about it by hand what's going to cancel everything from nine and down so what will I have on top 13 * 12 * 11 times 10 We're not gonna have anything else because nine and down cancels out what else am I going to have on the bottom all these nines cancel out nine factorial cancels out so what's left four * 3 * 2 * 1 so that's what you're going to be working out what will all of that equal I hear 715 715 okay make sure you can get that and if you're going to put it in the calculator like this up top don't forget you're going to have to have an extra set of parentheses around the denominator okay so try it make sure you can get it put your calculator try and make sure you can get it all right how many want to see a shortcut all right so what I would like for you to do is grab your calculators and I just want you to go over to math and go over to probability and do you see options two and three two would be your permutation and three would be your combination so let's just check this one okay first things first you you got to put in the 13 first so clear out your screen blank screen I want you to put in 13 everybody put in 13 first all I've asked you to do is put in 13 now I want you to go to math and go over to probability and we're going to choose Choice number three and then after that we're going to do our rate of four so in your calculator you should see 13 NCR four and hit enter what do you get 715 okay you like it so let's try a couple others let's look at example four let's put in now this time we're going to do what's our rate whoops so we have seven is our number it's a permutation right and our rate is seven at a time so we're going to do seven math probability Choice two this time and seven at a time so your n is seven your rate is seven click enter what do you get 5,40 which is what we got when we did it by hand okay let's take a look at this one this one is combination or permutation so we're going to do seven combination with a rate of two at a time so seven math probability [Music] combination two at a [Laughter] time okay any questions all right so