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Combination example: 9 card hands

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

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  • leaf green style avatar for user Chris Hendrickson
    What about the suits? Do they have any relevance in this? Would we multiply the final result shown by 4?
    (27 votes)
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  • leaf grey style avatar for user rprose
    and then if i do care about the order, i just don't add the extra k! in the denominator, right?
    (23 votes)
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    • leaf blue style avatar for user Matthew Daly
      That's right. Notice that if you had k people out of order and you cared about ordering them, you could choose the first person in k ways, the second person in k-1 ways, and all the way down to the last person being the only person left, so it works out that there were k! ways to sort them.
      (21 votes)
  • leafers seedling style avatar for user patch
    Say a word has multiple letters repeating- like ARCHITECTURE- the R, C,E and T repeat. The practice questions only have one letter repeating like PLOP (P) or ERROR (R). How would you compute the number of distinct permutations you can get from the word ARCHITECTURE?
    (11 votes)
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    • male robot johnny style avatar for user Thomas B
      You just need to divide by the number of combinations for each duplicate letter.

      ARCHITECTURE has 12 letters with 2 Rs, 2 Cs, 2 Ts, and 2 Es so you have 12!/(2!*2!*2!*2!) = 29,937,600 permutations

      Another example would be MISSISSIPPI, 11 letters with 4 Ss, 4 Is, and 2 Ps and 11!/(4!*4!*2!) = 34,650 arrangements.
      (19 votes)
  • blobby green style avatar for user Kyle Jackson
    Why is it not 36P9?
    (11 votes)
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    • blobby green style avatar for user vivek
      in 36P9 we would count a hand if the same cards in different order. however whatever order the cards come in, if they are the same, it will be the same hand. we would not want to count the same hand twice. hence order wont matter. so, we use 36C9. ie. we divide 36P9 with the number of different ways 9 things can be ordered (9!). hope i am right.
      (10 votes)
  • blobby green style avatar for user 😊
    how can i know that a particular question is permutation or combination
    (6 votes)
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    • duskpin ultimate style avatar for user Polina Vitić
      In permutations, the order of the results matters.
      - license plate number
      - lottery number
      - serial number
      - phone number
      - safe code
      - password

      In combinations, the order does not matter.
      - card games
      - nominees for government office
      - pizza toppings

      In order to tell the difference, just ask yourself if the order of the results matters:
      Yes? = permutation
      No? = combination

      Hope this helps!
      (14 votes)
  • male robot hal style avatar for user Charlie Norris
    What if the denominator became 0 factorial?
    (5 votes)
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  • piceratops ultimate style avatar for user Jimmy
    Can someone please prove why the permutation formula and the combinations formula? As Sal said, DON'T MEMORIZE THE FORMULAS!
    (1 vote)
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    • duskpin ultimate style avatar for user TotallyNotAFurryÒwÓ
      Sure, I'd be happy to provide an explanation for the permutation formula and the combinations formula.

      Permutations:
      A permutation is a way of arranging objects in a particular order. The number of ways to arrange n objects in a specific order is given by the formula:

      n! / (n-r)!

      where n is the total number of objects and r is the number of objects being arranged.

      The reasoning behind this formula can be understood as follows:

      For the first object, we have n choices.
      For the second object, we have n-1 choices (since we've already picked one).
      For the third object, we have n-2 choices (since we've already picked two).
      This continues until we have r objects picked, at which point we have n-r+1 choices.
      So the total number of arrangements is n * (n-1) * (n-2) * ... * (n-r+1).
      But this formula counts each arrangement r! times, since we could arrange the r objects in any order.
      So we need to divide by r! to get the final answer: n! / (n-r)!.
      Combinations:
      A combination is a way of selecting objects from a larger set without regard to their order. The number of ways to choose r objects from a set of n objects is given by the formula:

      n! / (r! * (n-r)!)

      The reasoning behind this formula can be understood as follows:

      To choose r objects from a set of n objects, we could first select any r objects (which can be done in n choose r ways).
      But this counts each combination r! times, since we could arrange the r objects in any order.
      So we need to divide by r! to correct for this overcounting.
      Also, we don't care about the order in which we choose the objects, so we need to divide by the number of ways to arrange the chosen objects, which is r!.
      So the final formula is n! / (r! * (n-r)!).
      I hope this explanation helps you understand the reasoning behind these formulas, rather than just memorizing them.
      (12 votes)
  • mr pink red style avatar for user limfxoo
    Why does dividing 36! by 9! make sense? I still don't understand why this formula give us the combination?
    Sorry, I always ask these silly questions... T_T
    (3 votes)
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    • leaf grey style avatar for user Nabla ∇
      What Sal did can be written as: 36!/(27! * 9!)

      Where:

      - 36! is the number of ways 36 cards can be arranged.
      - 27! is the number of ways the remaining 36 - 9 = 27 cards can be arranged. It makes sense, since you don't care about the arrangement of the cards you are not going to have in a 9-card hand.
      - 9! is just the number of ways you can arrange your hand after picking the 9 cards. It makes sense, because if I gave you 9 cards, you can arrange them however you want (arrange them by color, by type of suit, correlatively, etc.), but all those arrangement you can make don't change the fact that they are still the same 9 cards I gave you before.

      Permutations care about those arrangements you can make after receiving the cards (i.e, order matters), while combinations don't.
      (6 votes)
  • marcimus purple style avatar for user arwentelian
    This keeps coming up in the practice, and while I can answer it now due to repetition, I don't understand it. is there a formula (my brain likes formulas)? You need to put your reindeer, Jebediah, Lancer, Ezekiel, Gloopin, and Bloopin, in a single-file line to pull your sleigh. However, Lancer and Ezekiel are best friends, so you have to put them next to each other, or they won't fly.
    How many ways can you arrange your reindeer?
    (4 votes)
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    • duskpin ultimate style avatar for user Polina Vitić
      Think of the "best friends" as one unit/reindeer, then figure out how many ways there are to arrange four reindeer in a single-file line.

      Here's the trick: remember how we pretended the two "best friends" were one reindeer unit? In each of those arrangements, either Lancer or Ezekiel can come first! And this gives you twice as many arrangements as you figured out in the first step.

      So, multiply the number of arrangements x2 to get the final total.

      Hope this helps!
      (2 votes)
  • blobby green style avatar for user joe jackson
    how do I calculate the number unique combinations if we have 4 letters (A,B,C and D) where AB is equal BA (similar to handshaking) and each letter by itself is a combination. The following shows all the acceptable combinations:
    A
    AB
    ABC
    ABCD
    AC
    ACD
    BC
    BCD
    BD
    C
    CD
    D
    (3 votes)
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    • leaf blue style avatar for user Dr C
      Off-hand I don't recall any formula to do this. We basically just have to repeat the process for each number of letters that we're considering. So we'd say:

      1 letter: 4 nCr 1 = 4
      2 letters: 4 nCr 2 = 6
      3 letters: 4 nCr 3 = 4
      4 letters: 4 nCr 4 = 1
      (5 votes)

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.