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.

you just confused the hell out of me.. i need to watch this again.

this is good. I hope i can do my exam tmr. Thanks khanacademy

I have a data exam tommorrow and your helping a lot 😀 I would like to know if it was possible if on a website or something you could have like practice questions because i know how to do it i just need practice with more difficult questions

nice

REPLACE SCHOOLS WITH YOUTUBE ! Lol =))

I'm not sure how "…can be sorted however the player chooses," indicates that the order doesn't matter.

this is the worst unitt!!!!1 i hate it

I know that the formula works, but I still have trouble grasping the logic of dividing by 9!.

Maybe one of you folks could help me out here.

TY!!!!!!

this is amazing thanks

There is already a botton to calculate factorial called n!, and you only needed to write this en the calculator: 36!/(27!*9!)

Notice that 36! over 27! give as the product of 36*35*34…*28.

36C9

THANK YOUUUUUU!!!

I'm confused, in the previous video the same concept the two questions have but every question answered in different way which doesn't fit one another!

why do you make this complicated? it could just be 36C9

This guy doesn't know nothing

honestly this is never going to be used in life. realistically when the fuck would anyone ever need this information.

Thank you 🙂 I like the way you explain it's so easy to understand.

i love you

Thanks for saving me from the mindfuck that's been devouring my mind for the past few hours.

If you play this at half speed he sounds hella stoned 🙂 i love these videos, they're super helpful!

Thank you so much. The monkeys in my brain is chomping at the following qustions

1. In this scenario what if you want to know the number of unique numbered cards you are holding like for example on a hand containing a unique deck without repeating numbers (7 of spades and 7 of diamonds would count as the same)

2. In an example of persons and chairs what if I have 5 chairs and 2 people and count the various ways these two people can be arranged, I belive i can just plug chairs into n and people into r) correct ?

i dont under stand how to actually solve it you gave a map but not show me the end

plz salman can u show some difficult examples

That explanation at the end has blowed my mind off! Your approach helped me a lot, thanks.

Can’t you also just subtract 27! From the numerator, then divide by 9! ?