 This bootcamp takes you through several mathematical concepts and techniques that you need in order to do the introduction to statistics course. In this module, we'll review permutations and combinations. Permutation is a particular arrangement. For example, if I ask you how many ways can you range letters A, B, and C, well, I'm listing all the permutations or arrangements. A, B, C is different from A, C, B, different arrangements. You can also do this B, A, C, or B, C, A, B, or C, B, A. So you really have six ways to arrange the letters A, B, C. Using a calculator, you'll see it'll be three permutation, three. Okay, now we know three permutation, three is six. Well, how do we know that? Okay, first N is how many objects, in this case letters, how many we have? So three letters, A, B, and C. Then we have R, which is going to be the number of slots, the three slots. Okay, so let's look at this. We have three letters, A, B, or C. So in the first slot, we have a choice, we put an A, a B, or a C. Suppose in slot one, we used an A. Now in slot two, we have a choice of B or C. Suppose we use the B for slot two. Well, that's left to C. So there's your first arrangement, A, B, C. If you start it with an A and then a slot two, you put a C. There's nothing left, except for a B, which goes to slot three, A, C, B. Suppose you start it with a B in slot one. Well, in slot two, you can put an A or a C. Let's see it started with an A in slot two, B, A. Now all that's left to C, you got B, A, C. If in slot two, you put the C after the B, so you have B, C. You must have an A that's all left. You got B, C, A. This is called a tree diagram, actually, if you do it, you'll see. So that's how we know that three permutation three ends up being three factorial. Three times two times one. So three permutation three is three factorial. If you look at your scientific calculator, you'll see an NPR key. P stands for permutations. N is the number of distinct objects before we had A, B, and C. So these are the objects you want to arrange, and R is the number of slots or spaces. So the previous example, we had three objects, A, B, C, to range in three slots, and that becomes three permutation three, which is three factorial, which is three times two times one, which is six. The general formula for permutation is NPR is N factorial divided by N minus R factorial. Thus, when N equals R, then NPN is just N factorial. And the way you read a factorial is showed to you, but just let's do with numbers. Ten factorial is ten times nine times eight times seven times six times five times four times three times two times one, which is more than three million, it's three million six hundred twenty eight eight hundred. So it's a large number. Six factorial is six times five times four times three times two times one or seven twenty. It's actually surprising how large these numbers become. You try 50 factorial, it's going to mean a credibly large number. Let's look at some problems. How many ways can you assign five workers to five different tasks? Well, that's five P five, which is five factorial, five times four times three times two times one, which is one twenty. Again, you have a factorial key, so you just get it directly. One twenty. How many ways, example two, how many ways can you arrange ten different books in your bookcase? And suppose it only has room for five books, that's like the slots, spaces. So we have ten permutation five, remember the second number is the slots. So it's ten permutation five and that turns out to be ten times nine times eight times seven times six. The five four three two one is cancelled out by the denominator. But if you just put in your calculator, you get thirty thousand two forty. Example three, how many ways can eight cars line up single file in front of a toll booth? You can resume this space for eight. So it's eight permutation eight because there are eight spaces and there are eight cars, which is eight factorial, which is forty thousand three hundred and twenty. The fourth example, how many ways can you arrange twelve guests around the table as twelve chairs? Well the chairs are not like the slots, so the spaces. So n is twelve, the twelve guests, you want to put them around the table as twelve chairs? That's twelve permutation twelve, which is twelve factorial. And look at the incredible number that is, four seventy nine, a million six hundred. And now you can see why so many family feuds occur when it comes to seating family members at weddings or bar mitzvahs or whatever family event you have. So many ways to arrange them and if your family is like my family, somebody is always going to say, why did you seat me next to Jane? You know I haven't talked to her in seven years, or why did you seat my daughter next to Ellen? You know they don't get along. Things like that. But look how hard it is, twelve permutation twelve is more than four hundred seventy nine million. Well now we're going to talk about permutations and combinations. Permutations, the arrangement of the items is important. Each unique sequence is another permutation, thus ABC is not the same as BCA, which is not the same as CBA, you just change around the arrangement the way it's ordered and it's a different permutation, so you get generally a larger number than you're looking at combinations. Combinations, ABC, BCA, CBA are not counted as three separate arrangements, it's the same combination. You still got ABC, the same three letters are in ABC, BCA, and CBA. So let's see how this works. So for example, if I ask you how many different groups of three can be selected from seven people? Let's call the people A, B, C, D, E, F, G, they're seven people. And now once you select B, D, and E, the fact that they can be arranged six different ways, B, D, E, E, D, B, E, D, it doesn't matter, it's only relevant, it's the same three. So obviously you can get a smaller number. So that's the difference between a permutation and a combination. Note you have both keys on your, you can do an NPR or NCR on your calculator. This is the formula for combination. N, C, not P, now it's a C, NCR is N factorial over R factorial times N minus R factorial. Notice it's almost the same as the permutation formula, except now you're dividing, you're shrinking it in fact by dividing it by R factorial. So that's why we see that NCR is NPR over R factorial. Anyway, you don't have to worry about all this, you have an NCR key on your calculator to solve any combination problem. Make sure you have a calculator that has that key. Okay, you shouldn't even buy a calculator that doesn't have NPR and NCR on it. They're very cheap now. So for example, how many different groups of three can be selected from seven people? Okay, that's really, sampling in general is a combination problem. So if you want to get groups of three from seven, again you don't care how they're ordered, once you get to three that's it. Okay, so seven, combination three, which is seven factorial divided by three factorial times four factorial, and if you put this into your calculator, you should get 35. How many different hands can one draw from a deck of 52 cards in a game of seven card rumbling? Again, you don't care about how it's ordered, you just want to see how many different decks, from a deck of 52, how many hands of seven can you get? Okay, that's 52, that's N, combination seven, that's like your slots, 52, combination seven, that's 52 factorial over seven factorial times 45 factorial. If you do it in your calculator, you'll find there's 133,784,560 different hands you can get in seven card rummy. Let's try the next example. How many samples of size six can be drawn from a population of size N equals 50? Well, this is simply 50 combination six, which is 50 factorial over six factorial times 44 factorial. Using your calculator, you'll get the answer 15,890,700, and that's, it's very simple once you learn how to use your calculator, you have no problem. The biggest problem you'll have is deciding, is it a permutation or a combination? And that's easy to figure out, do we care about the arrangement or not? If you don't care about arrangement and once you have ABC, it's the same as BCAC, then you're basically looking at a combination. To find more boot camp modules, visit the stat course at the URL you see there and go to the navigation bar on the left, click boot camp and you'll see all kinds of things that are good to do prior to the statistics course. Many of you have already done this before and maybe only need a refresher.