  Let $\mathcal{N}[k]$ be the multiset containing the $\binom{n-1}{k}$ products
of $k$-subsets of $\{1,\ldots, n-1\}$. We show that if $n\geq (2c+3)^2$, then
\begin{gather*}\left((-1)^c+\sum_{M\in
\mathcal{N}[n-1-c]}M\right)\cdot(c+1)\equiv 0\pmod{n},\end{gather*} if and only
if $n=(c+1)p$, where $p$ is prime. This provides a combinatorial extension of
Wilson's Theorem, which is the special case where $c=0$.
