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I have been struggling with the following problem, which I have been trying to solve combinatorially, but without much success.

Suppose n players each have a deck of cards. Each player randomly draws a hand of m cards from their own deck.

The easier part of the question is: what is the probability that there is (at least) one card which appears in every player's hand?

But I am really more interested in the harder part of the question: what is the probability that there is (at least) one set of k cards such that every player has one of the k cards in his hand?

For example, when $k=2$, what is the probability that we can find a pair of cards A and B such that every player has either or both of A and B among his hand of m cards?

Any advice on how to proceed or where this might have been previously covered would be much appreciated.

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    I found the equation to be correct, and I might be missing something, can you explain a situation where he "double counts".2012-06-07
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    Equation now removed - it was incorrect because it would count twice each arrangement of cards where all players had 2 out of _m_ cards in common. Plus of course higher numbers of shared cards up to the case where all players are holding the same _m_ cards.2012-06-07

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