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Problem

A certain casino uses 10 standard decks of cards mixed together into one big deck, which we will call a superdeck. Thus, the superdeck has 52 · 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 cards does not matter, nor does it matter which of the original 10 decks the cards came from. Express your answer as a binomial coefficient. Hint: Bose-Einstein.

My attempt at a solution

Because the number of each type of card is equal to the hand's size the number of cards is not limiting, so this can be thought of as a problem involving sampling with replacement.

There are 52 choices for each card and because the order doesn't matter this gives 52^10/10! combinations.

What is wrong about this reasoning?

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  • $\begingroup$ The hand that has all $10$ of the Aces of Spades doesn't have $10!$ distinguishable arrangements, only $1$. Similar issue with any hand that had repeat cards. $\endgroup$ Jul 30 '20 at 0:08
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One thing that is wrong is that $52^{10}/10!$ is not an integer

Another is that if for example, all ten cards are the Queen of Hearts then you should not be dividing by $10!$

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