Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 2 / 9F / a / Solution

Conditional on $N=n$, there are $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ pairs, and each is both friendly and birthday-matched with probability $1/(2\cdot 365)$. Since $\mathbb{E}[N(N-1)]=365^2$ for a Poisson variable,

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