Assume that there are $n$ balls numbered from $1,2,\ldots,n$ and $n+1$ urns, numbered as $0,1,\ldots,n$
Throw each ball randomly into one of $n$ urns: urn 1, urn 2, . . . , urn $n$. That is, any urn except urn $0$. (A ball goes to a certain urn with probability $1/n$) Check each urn and if there are more than one ball in an urn, choose one randomly and keep that in that urn and remove the other balls and put them into urn $0$.
What is the probability that there are $k$ balls in urn 0?