Although it seems elementary, I am not aware that standard textbooks treat this problem.
There is a universal set U of N distinct objects. A fixed subset S of n distinct objects is chosen from it (0 < n < N).
Another subset T of m (0 < m < N) distinct objects is then chosen from U. The question is what is the probability distribution of the cardinality of S intersection T. N may be considered to be infinity, although m/N and n/N are not vanishingly small.