Say that we have $N$ identical balls with $M$ bins and let $n_i$ denote the number of balls we know to be in bin i, $\sum^M_{i=1}n_i = N$.
How would one then calculate the probability of getting this particular configuration when the balls are equally likely to be in each bin and there are no restrictions on how many balls a bin can contain?
If we were to impose a condition that limited the number of balls a bin can contain, how would this translate to the problem?
If you imagine the balls are labeled, then each outcome has probability $2^{-N}$. You just need to multiply this by the number of outcomes that have the specified counts $n_i$, which is the "multinomial coefficient" $$\binom{N}{n_1,n_2,\ldots,n_M} := \frac{N!}{n_1! n_2! \cdots n_M!}.$$