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The inductive size bias coupling technique and Stein's method yield aBerry-Esseen theorem for the number of urns having occupancy $d \ge 2$ when $n$balls are uniformly distributed over $m$ urns. In particular, there exists aconstant $C$ depending only on $d$ such that $$ \sup_{z \in\mathbb{R}}|P(W_{n,m} \le z) -P(Z \le z)| \le C \left(\frac{1+(\frac{n}{m})^3}{\sigma_{n,m}} \right) \quad \mbox{for all $n \ge d$and $m \ge 2$,} $$ where $W_{n,m}$ and $\sigma_{n,m}^2$ are the standardizedcount and variance, respectively, of the number of urns with $d$ balls, and $Z$is a standard normal random variable. Asymptotically, the bound is optimal upto constants if $n$ and $m$ tend to infinity together in a way such that $n/m$stays bounded.

A Berry-Esseen bound for the uniform multinomial occupancy model