# Probability, ball and bin with constraint

Assume that we have $n$ bins(indistinguishable and numbered from $1$ to $n$). We pick a bin and add a ball to it subject to the constraint that each bin can have no more than $n$ balls.

If we know that after iterating this process for a while, there is a total of $m$ ball in the bins, then can we say something about the probability that for example the first bin contains $m_1$ balls, the second $m_2$ and etc? in other words: $\text{Pr} [(m_1,m_2,\cdots,m_n)]$.

Thank you!

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This problem looks quite tough, I was wondering where did you find it? – Lord Soth Jul 5 '13 at 22:50
This is what I came up while solving another problem in randomized gossip algorithms and need to have the answer or some bounds for the above probability to solve the original problem. – shirin Jul 5 '13 at 22:54