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## Precalculus

# Combination example: 9 card hands

CCSS.Math:

Thinking about how many ways we can construct a hand of 9 cards. Created by Sal Khan and Monterey Institute for Technology and Education.

## Video transcript

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.