Expected and Variance of Distinct Birthday Count

Among $n$ people whose birthdays are independent and uniform on $\{1, \ldots, d\}$, let $D$ be the number of distinct birthdays observed.

(a) Derive $E[D]$ using indicator random variables.

(b) Derive $\operatorname{Var}(D)$. You will need $P(\text{day } j \text{ and day } k \text{ both occupied})$ for $j \ne k$.

(c) For $n = 100$ and $d = 365$, compute $E[D]$, $\operatorname{Var}(D)$, and the expected number of "collision people" $n - D$ (people whose birthday coincides with at least one other person).

(d) Is $E[n - D]$ the same as the expected number of collision pairs $\binom{n}{2}/d$ from the indicator-pair approach? Explain the distinction.