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.