Variance of Birthday-Collision Pair Count

Continuing from the setup of the expected collision-pair count: $n$ people have independent uniform birthdays on $\{1,\ldots,d\}$. Define $X = \sum_{i<j} \mathbf{1}[B_i = B_j]$.

(a) Compute $\operatorname{Var}(X)$.

(b) A surprising intermediate step: show that $\operatorname{Cov}(\mathbf{1}[B_i = B_j],\, \mathbf{1}[B_j = B_k]) = 0$ for distinct $i,j,k$ even though the two indicators share the index $j$. Explain intuitively why this zero covariance holds.

(c) For $d = 365$ and $n = 28$, compute $\operatorname{Var}(X)$ numerically and give the coefficient of variation $\sigma_X / E[X]$.