Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}, let $\mathbb{N}$ be the set of \reftext{def:natural-numbers-2026a}{natural numbers} with $\mathbb{N}_0=\mathbb{N}\cup\{0\}$, and let $n,m\in\mathbb{N}$. Let $V_1,\dots,V_n$ be \reftext{def:independence-events-rvs-2026a}{independent} random variables, each with the same \reftext{def:distribution-cdf-random-variable-2026a}{distribution} $\nu$. Let $A_1,\dots,A_m$ be pairwise disjoint \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel sets} whose union is $\mathbb{R}$, the set of \reftext{def:real-numbers-c54-2026c}{real numbers}, and set $p_j=\nu(A_j)$ for $1\le j\le m$.

For $1\le j\le m$ define the \textbf{cell count}

where $\mathbf{1}_{E}$ denotes the function equal to $1$ on the event $E$ and $0$ on its \reftext{def:complement-subset-relative-set-2026a}{complement}. Each $S_j$ is a random variable, as shown in the proof.

Then for every $(n_1,\dots,n_m)\in\mathbb{N}_0^{m}$ with $n_1+\dots+n_m=n$,

with the \reftext{def:factorial-natural-number-2026a}{factorial} (convention $0!=1$), the convention $x^{0}=1$, and the \reftext{def:finite-product-notation-2026a}{finite product notation}.