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We analyze the asymptotic behavior of the average maximal number of balls ina bin obtained by throwing uniformly at random $r$ balls without replacementinto $n$ bins, $T$ times. Writing the expected maximum as $\frac{r}{n}T+C_{n,r}\sqrt{T} + o(\sqrt{T})$, a recent preprint of Behrouzi-Far andZeilberger asks for an explicit expression for $C_{n,r}$ in terms of $n,r$ and$\pi$. In this short note, we find an expression for $C_{n,r}$ in terms of $n,r$ and the expected maximum of $n$ independent standard Gaussians. Thisprovides asymptotics for large $n$ as well as closed forms for small $n$---e.g.$C_{4,2} = \frac{3}{2 \pi^{3/2}} \arccos(-1/3)$---and shows that computing aclosed form for $C_{n,r}$ is precisely as hard as the difficult question offinding the expected maximum of $n$ independent standard Gaussians.

A Short Note on the Average Maximal Number of Balls in a Bin