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A certain casino uses 10 standard decks of cards mixed together into one big deck, which we will call a superdeck. Thus, a superdeck has $52{\cdot}10=520$ cards, with 10 copies of each card. How many different 10-card hands can be dealt from the superdeck? The order of the hands does not matter, nor does it matter which of the original 10 decks the cards came from. Express your answer as a binomial coefficient.

My attempt:

This amounts to sampling from the set $\{1,2,3,\ldots,52\}$, $10$ times with replacement and without ordering.

This problem is similar to having $10$ indistinguishable particles (cards) to be put into $n=52$ distinct boxes. The particles are indistinguishable - that means we do not care about the order in which the cards are chosen.

{Ace, Two, Three, Four, Five, Six, Seven, Eight, Nine, Nine} of hearts is treated identical to {Nine, Nine, Eight, Seven, Six, Five, Four, Three, Two, Ace}.

Only the counts for how many particles in each box - that is how may aces of heart, two's of a club etc. matter.

The number of ways to do this is given by the Bose-Einstein result

${n+k-1\choose{k}}={61\choose10}$ different 10-card hands.

(a) Is my solution to the problem correct? (b) I want to try and list enumerate the different possibilities and count them. I am not able to count them individually. Could someone help me in that...

For example, we are interested in the different possible 10-card hands like,

  • All 10 cards are the same. (XXXXXXXXXX) ${52\choose1}$
  • 9 cards are the same, 1 card is different (XXXXXXXXXY). ${52\choose1}\cdot{51\choose1}$
  • 8 cards are the same, 2 cards are different.
  • 8 cards are the same, 2 other cards are also the same.
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  • $\begingroup$ See math.stackexchange.com/questions/1054808. $\endgroup$ – joriki Jul 18 '16 at 10:05
  • $\begingroup$ @joriki, Thanks for pointing to that amazing insight behind the Bose-Einstein result. As an exercise, I wanted to try and enumerate all of the different possibilities. Would it be possible to do that? $\endgroup$ – Quasar Jul 18 '16 at 11:41

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