We randomly place $n$ balls in $m$ bins independently and uniformly. For each $i$ with $1 \leqslant i \leqslant m$, let $B_{i}$ be the number of balls in bin $i$.

(a) What is the distribution of $B_{i}$ ? For $i \neq j$, are $B_{i}$ and $B_{j}$ independent?

(b) Let $E$ be the number of empty bins, $C$ the number of bins with two or more balls, and $S$ the number of bins with exactly one ball. What are the expectations of $E, C$ and $S$ ?

(c) Let $m=a n$, for an integer $a \geqslant 2$. What is $\mathbb{P}(E=0)$ ? What is the limit of $\mathbb{E}[E] / m$ when $n \rightarrow \infty$ ?

(d) Instead, let $n=d m$, for an integer $d \geqslant 2$. What is $\mathbb{P}(C=0)$ ? What is the limit of $\mathbb{E}[C] / m$ when $n \rightarrow \infty$ ?