The first part of the question (Finding expected number of distinct values selected from a set of integers) has been answered a few times.
Can someone provide a solution if we're drawing from a binomial distribution instead of a uniform distribution?
In other words, let X1,X2,X3,…,Xk be independent Binomial (n,p) random variables. What is the expected number of distinct values in the set {X1,…,Xk}?