#StackBounty: #combinatorics What is the distribution of the cardinality of the intersection of independent random samples without repl…

Bounty: 50

$S$ is some set with $ninmathbb{N}$ elements, and $a_1,a_2,…,a_m$ are fixed positive integers less than or equal to $n$.

With the elements of $S$ being equally likely, $m$ samples $L_1, L_2,…,L_m$ are separately and independently drawn from $S$ without replacement, the size of which are $a_1,a_2,…,a_m$, respectively.

The cardinality of the intersection of the samples $left|L_1cap L_2cap … cap L_mright|$ has, in general, support equal to ${0,1,…,min{a_1,a_2,…,a_m}}$, but which distribution does it follow?

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