balls and bins - the probability of reaching the exact expected value

Question

Suppose I throw $n$ balls to $m$ bins (randomly and independently). The expected value of number of balls in each bin is $n/m$. How can I tell (or bound) the probability that every bin has exactly $n/m$ balls in it?

Answer 1

Assume that $n=mk$ for some positive integer $k$, otherwise the event in consideration is impossible. Then the distribution of the collection of numbers of balls in each of the $m$ bins is multinomial with parameters $\left(n;\frac1m,\frac1m,\ldots,\frac1m\right)$, where the probability $\frac1m$ is repeated $m$ times, to sum to $1$.

Hence the probability $p$ that it is exactly $(k,k,\ldots,k)$ (the number $k$ being repeated $m$ times, to sum to $n$) is $$ p={n\choose k,k,\ldots,k}\left(\frac1m\right)^k\,\left(\frac1m\right)^k\cdots\left(\frac1m\right)^k=\frac{n!}{(k!)^mm^n}. $$ For large values of $n$ and/or $m$ and/or $k$, Stirling formula yields precise equivalents of $p$. For example, if $k\to\infty$, $$ p\sim\sqrt{\frac{m}{(2\pi k)^{m-1}}}. $$