While exploring ways to estimate multiset union and intersection cardinalities, I discovered an interesting result mentioned in a paper by Shukla , who in turn cites Feller : If r elements are chosen uniformly and at random from a set of n elements, the expected number of distinct elements obtained is
Given an element a from the set, the probability that there are none of the elements after r draws is a is . Thus the probability that there is at least one element is a (or equivalently that there is at least one unique item) is , and hence the expected number of uniques is . Alternatively, as Feller explains, the probability of a single unique at the ith step is , hence the expected number of uniques after r draws is
A geometric series that converges to .
 A. Shukla et al, Storage estimation for multidimensional aggregates in the presence of hierarchies [paper]
 W. Feller, An introduction to probability theory and its applications, vol. 1, 1957