= Solution
For any fixed bin, each ball supplies an independent <Bernoulli distribution> indicator with success probability $1/m$. Summing the $n$ <indicator variables> gives
$$
\boxed{B_i\sim\operatorname{Bin}(n,1/m).}
$$
The full occupancy vector has a <multinomial distribution>, the basic <uniform balls-in-bins allocation>. Distinct bin counts are \b[not independent] when $n\geq1$ and $m\geq2$. To see this, write $B_i=\sum_{r=1}^n I_{r,i}$. For the same ball, $I_{r,i}I_{r,j}=0$ if $i\ne j$, while indicators from distinct balls are independent. Their <covariance> is therefore
$$
\boxed{\operatorname{Cov}(B_i,B_j)=-\frac{n}{m^2}<0\quad(i\ne j).}
$$
With no balls the counts are deterministic; with one bin there are no distinct bin pairs. These degenerate cases do not contradict the conclusion.
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