Given is a sequence $\langle a_1,a_2,\ldots,a_n\rangle$ over the alphabet $\{1,2,\ldots,m\}$ chosen uniformly at random among the $m^n$ possibilities. What is the expected size of the set $\{a_1,a_2,\ldots,a_n\}$?
If $m=n$ it seems the answer tends to $(1-1/e)n$ as $n\to\infty$, but I don't know why.
I bumped into this while benchmarking some code for hashtables, so I wouldn't be surprised if it is a standard result in the hash world.