# $m$ balls into $n$ already occupied urns

Assume that there are n urns, $k\in\{1,...,n-1\}$ of which have already at least one ball. $m$ new balls will be thrown into urns. Each of the $m$ balls is thrown randomly and uniformly into $n$ urns. That is, each ball goes into each urn with probability $1/n$.

What is the probability that there are then exactly $r$ non-empty urns among those urns which were not occupied at first?

( assume that both balls and urns are distinguishable)

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To get $\{$ and $\}$, use \{ and \}. –  Brian M. Scott Jun 5 '13 at 17:25