 In this lecture, we review some basic high school math concepts that you'll need in our statistics course. Before we get to the binomial distribution, we have to take a slight regression and review permutations and combinations. A permutation is an arrangement. So, for example, I ask you how many ways can you arrange the letters A, B, and C? Note that ABC, ACB, BAC, BCA, CAB, and CBA have six different permutations of the letters A, B, and C. Of course, they all have the same, in quotes, combination. They all have A, B, and C in them. But as a permutation, since it's arranged differently, ABC is not CBA. It's a different arrangement. So, we're going to learn how to do a permutation. This is a formula for that. Well, we have three letters, A, B, and C. We want you to know how many ways, how many permutations you have. Well, we have to know how many slots you have, positions. All right? So, we're going to define N, how many objects, get A, B, and C, and we have R, the number of slots. So, slot number one, we can put an A, B, or C in. And slot two, once you start with, let's say, slot when you have an A, there's only two things left for slot two. So, you start with A, you've got B, C, B, or C. And then, of course, there's only one thing left for slot three. So, if you start with an A, and you put a B in slot two, then slot three, or you're left with a C. Now, let's say you start in slot one, you start with a B, right? In slot two, you've got a choice, A or C. And let's say you have B, then A, then you're stuck. In slot three, you've got only one. So, again, in slot one, we have three letters to choose from, A, B, or C. In slot two, you're only going to have two letters left. And finally, in slot three, you only have one unused letter. Notice it's three times two times one. That's how we got the six. And you can see the six different permutations. A, B, C, A, C, B, B, A, C, A, C, A, B, C, B, A. When you do this in your calculator, you'll see a key that has N, P, R. The P stands for permutations. Usually it's together with the N, C, R key. One will be the first function, one will be the second function. So, both be on your calculator. It's the N, P, R, and N, C, R. It's got to be a scientific calculator. So, P stands for permutation. N is the number of distinct objects you want to arrange. And R is the number of slots or spaces, if you wish. So, the previous example, we started with A, B, C. We have three objects, A, B, C. We want to put them into three slots. On your calculator, that will be three permutation, three. Three P, three. And when you play with it in your calculator, you'll see it's six. Three P, three is six. The general formula for permutation, you want to do it by formula, not by calculator, NPR is N factorial over N minus R factorial. Okay? What does factorial mean? So, 10 factorial means 10 times 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. This is a huge number. Alright? 6 factorial is 6 times 5 times 4 times 3 times 2 times 1, which actually is 720. Always remember, 0 factorial is 1. Okay? So now, so NPR is N factorial over N minus R factorial. Of course, when N equals R, the NPM becomes N factorial over N minus N, which is 0 factorial, which is 1. So NPN is simply just N factorial. Okay? So this is a formula that you can use if you want to use a formula. So NPR is N factorial over N minus R factorial. And remember that if R equals N, then you're just looking at N factorial over 1, or just N factorial. Let's look at some problems that involve permutations. How many ways can you assign 5 workers to 5 different tasks? Well, N is 5 if 5 workers. The task for your slots. Okay? So you have 5 P5. Well, here's where N equals R. So it's 5 factorial over, if you want, over 0 factorial, which is 1. So it's just 5 factorial. This is 5 times 4 times 3 times 2 times 1. That's 120. So there are 120 ways you can do this. How many ways... Example 2. How many ways can you arrange 10 different books in your bookcase? And you have a bookcase, a small one. And it's ruled for exactly 5 books. Well, that's 10. N is 10. 10 books. But you only have 5 slots. So 10 P5. And that works out to 10 times 9 times 8 times 7 times 6. Works out to 30,240. You really should do this in your calculator. It's just 10 P5. How many ways can 8 cars line up single file in front of a total booth? Well, again, here we have 8 cars. N is 8. But we have 8 slots. There are 8 positions for these cars. So it's 8 P8, which is 8 factorial, which is 40,020. Here's another one. How many ways can you arrange 12 guests around the table and have 12 chairs? The chairs are like slots. Well, you have 12 guests. And it's 12 P12 or 12 factorial ways of arranging them. And that works out to 479,600,000. And as we note, this is why you get lots of fights in families with weddings and seating arrangements, because they say something like, you know, Jane O'Rourke, why'd you seat me near John? You know, I don't like John. You know, it's a permutation problem. And every table, and you can have 12 tables if there's an issue. With permutations, the arrangement is important. Each unique sequence is another permutation. As I mentioned, ABC is not the same as DCA. And they're both different from CBA. It's like the thing we have in the seating plan. John near Jane is not the same as pudding Jane near John, near somebody else. You know, each one's a different arrangement. With combinations, on the other hand, ABC, BCA, CBA, it's all the same thing. We all have the same combination of A, B, and C in them. So, for example, I ask you, how many different groups of three can be selected from seven people? So we're going to call these people ABC, D, E, F, G, we have seven people, and we want how many different groups of three? Of course, once you select B, D, and E, the six different ways you can arrange B, D, and E is irrelevant. It's full B, D, and E. It's kind of with cards, of course. You have a hand of five cards, and you have four aces. Nobody cares on how they were arranged. You can say, I've got four aces in my hand. On your calculator, you'll see NCR. It's basically NPR. But you've got to shrink it, reduce it by R factorial. So the formula for NCR, N combination R, N objects, R slots, NCR is N factorial. Over R factorial times N minus R factorial. Again, use your calculator. That's the best way to do this. So the previous problem, we had how many groups of three can be selected from seven people? That's seven combination three. Well, if you do any calculator, you get the answer 35. If you want to do it by hand, it's seven factorial over three factorial times four factorial. A little trick. In the denominator, the two numbers always add up to the numerator. It's got to add up to N. So when you see three and four, you don't answer the seven, so you probably got to write. Okay, so just use your NCR key. Once you recognize that a problem is a combination problem. If it's a permutation problem, you're going to use the NPR key for a combination problem and NCR key. Look at this problem. Example one. How many different hands can one draw from a deck of 52 cards? That's a typical deck. And you're playing seven card Rami. So how many hands of seven? Remember, this is a combination. Because we don't really care about how it comes in. You just want to know how many different hands can you get? Well, 52 combination seven. Notice the 52 factorial over seven factorial times 45 factorial. Again, seven and 45 is 52. See how you got it right. 52 combination seven is a huge number. One and 33 million, seven, 84, five, 60. That's your answer. Example two. How many samples of size N equals six can be drawn from a population of size capital N equals 50? Notice using small N for the sample. The population now, capital N is 50. Well, that's 50 combination six. And notice how large that is. 15 million, 890, 700. Quite a few different possible samples that you can get. In fact, samples in general should be seen as a combination situation, a combination problem.