Summary: Consider a fair $n$-sided die with faces numbered 1 to $n$. Several different methods are used to compute the probability that every face has come up at least once when face $n$ appears for the $k^{th}$ time. The results lead to a number of summation identities. The probabilities are related to several sequences in Sloane's On-Line Encyclopedia of Integer Sequences.

11B99, 60C05

occupancy problem, waiting time, inclusion-exclusion counting, Stirling number of the second kind, Markov chain, binomial identity, binomial sum, multinomial sum, binomial transformation