Thinking about how many ways we can construct a hand of 9 cards. Created by Sal Khan and Monterey Institute for Technology and Education.
A card game using 36 unique cards, four suits, diamonds, hearts, clubs and spades-- this should be spades, not spaces-- with cards numbered from 1 to 9 in each suit. A hand is chosen. A hand is a collection of 9 cards, which can be sorted however the player chooses. Fair enough. How many 9 card hands are possible? So let's think about it. There are 36 unique cards-- and I won't worry about, you know, there's nine numbers in each suit, and there are four suits, 4 times 9 is 36. But let's just think of the cards as being 1 through 36, and we're going to pick nine of them. So at first we'll say, well look, I have nine slots in my hand, right? 1, 2, 3, 4, 5, 6, 7, 8, 9. Right? I'm going to pick nine cards for my hand. And so for the very first card, how many possible cards can I pick from? Well, there's 36 unique cards, so for that first slot, there's 36. But then that's now part of my hand. Now for the second slot, how many will there be left to pick from? Well, I've already picked one, so there will only be 35 to pick from. And then for the third slot, 34, and then it just keeps going. Then 33 to pick from, 32, 31, 30, 29, and 28. So you might want to say that there are 36 times 35, times 34, times 33, times 32, times 31, times 30, times 29, times 28 possible hands. Now, this would be true if order mattered. This would be true if I have card 15 here. Maybe I have a-- let me put it here-- maybe I have a 9 of spades here, and then I have a bunch of cards. And maybe I have-- and that's one hand. And then I have another. So then I have cards one, two, three, four, five, six, seven, eight. I have eight other cards. Or maybe another hand is I have the eight cards, 1, 2, 3, 4, 5, 6, 7, 8, and then I have the 9 of spades. If we were thinking of these as two different hands, because we have the exact same cards, but they're in different order, then what I just calculated would make a lot of sense, because we did it based on order. But they're telling us that the cards can be sorted however the player chooses, so order doesn't matter. So we're overcounting. We're counting all of the different ways that the same number of cards can be arranged. So in order to not overcount, we have to divide this by the ways in which nine cards can be rearranged. So we have to divide this by the way nine cards can be rearranged. So how many ways can nine cards be rearranged? If I have nine cards and I'm going to pick one of nine to be in the first slot, well, that means I have 9 ways to put something in the first slot. Then in the second slot, I have 8 ways of putting a card in the second slot, because I took one to put it in the first, so I have 8 left. Then 7, then 6, then 5, then 4, then 3, then 2, then 1. That last slot, there's only going to be 1 card left to put in it. So this number right here, where you take 9 times 8, times 7, times 6, times 5, times 4, times 3, times 2, times 1, or 9-- you start with 9 and then you multiply it by every number less than 9. Every, I guess we could say, natural number less than 9. This is called 9 factorial, and you express it as an exclamation mark. So if we want to think about all of the different ways that we can have all of the different combinations for hands, this is the number of hands if we cared about the order, but then we want to divide by the number of ways we can order things so that we don't overcount. And this will be an answer and this will be the correct answer. Now this is a super, super duper large number. Let's figure out how large of a number this is. We have 36-- let me scroll to the left a little bit-- 36 times 35, times 34, times 33, times 32, times 31, times 30, times 29, times 28, divided by 9. Well, I can do it this way. I can put a parentheses-- divided by parentheses, 9 times 8, times 7, times 6, times 5, times 4, times 3, times 2, times 1. Now, hopefully the calculator can handle this. And it gave us this number, 94,143,280. Let me put this on the side, so I can read it. So this number right here gives us 94,143,280. So that's the answer for this problem. That there are 94,143,280 possible 9 card hands in this situation. Now, we kind of just worked through it. We reasoned our way through it. There is a formula for this that does essentially the exact same thing. And the way that people denote this formula is to say, look, we have 36 things and we are going to choose 9 of them. Right? And we don't care about order, so sometimes it'll be written as n choose k. Let me write it this way. So what did we do here? We have 36 things. We chose 9. So this numerator over here, this was 36 factorial. But 36 factorial would go all the way down to 27, 26, 25. It would just keep going. But we stopped only nine away from 36. So this is 36 factorial, so this part right here, that part right there, is not just 36 factorial. It's 36 factorial divided by 36, minus 9 factorial. What is 36 minus 9? It's 27. So 27 factorial-- so let's think about this-- 36 factorial, it'd be 36 times 35, you keep going all the way, times 28 times 27, going all the way down to 1. That is 36 factorial. Now what is 36 minus 9 factorial, that's 27 factorial. So if you divide by 27 factorial, 27 factorial is 27 times 26, all the way down to 1. Well, this and this are the exact same thing. This is 27 times 26, so that and that would cancel out. So if you do 36 divided by 36, minus 9 factorial, you just get the first, the largest nine terms of 36 factorial, which is exactly what we have over there. So that is that. And then we divided it by 9 factorial. And this right here is called 36 choose 9. And sometimes you'll see this formula written like this, n choose k. And they'll write the formula as equal to n factorial over n minus k factorial, and also in the denominator, k factorial. And this is a general formula that if you have n things, and you want to find out all of the possible ways you can pick k things from those n things, and you don't care about the order. All you care is about which k things you picked, you don't care about the order in which you picked those k things. So that's what we did here.