Example: 9 card hands Thinking about how many ways we can construct a hand of 9 cards
Example: 9 card hands
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- 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.
- 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
- 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.
- 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.
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