How many of the arrangements have a first duplication on sample number i? There are N ways to choose the ith sample. This element also appears exactly once in samples 1 through i-1, and disregarding order, there are possible choices for the other elements. Each set of elements in samples 1 through i-1 can be ordered in ways. So altogether, there are
possible sequences of samples which have a first match on the ith sample.
So the probability that a first match happens on sample number i is
and the expected number of samples required to achieve a match is