Lets say we have $N$ objects and $C$ bins. Let $B_k$ be the number of objects in bin $k$ after we distributed each of the $N$ objecs into the $C$ bins. Let us assume that for each object we made a random decision where we chose each bin equally likely. So I figured out that $B_k$ must have a Bernoulli distribution with parameter $p = 1/C$, but what is the distribution of the multivariate random variable $B = (B_1,B_2,\ldots,B_C)$?
Obviously the $B_i$ are not independent as $B_1 + B_2 + \ldots + B_C = N$.
Assume that objects are placed independently in the bins. Then $B_k \sim Binomial(N, \frac{1}{C})$ and $$P(B_1=n_1, \ldots, B_C=n_C) = {N\choose n_1,\ldots, n_C} \frac{1}{C^N}$$ when $n_1+\cdots+n_C=N$.