Suppose we have a family $\mathfrak{A}$ of some subsets of $\Omega$, which is locally finite, i.e. $$ X(\omega): = \sum_{A \in \mathfrak{A}} \mathbf{1}_{A}(\omega) <\infty $$ for all $\omega\in \Omega$.
Then for any $k\ge 1$, $$ (X)_l = X(X-1)\cdots(X-k+1) = \sum_{\text{distinct } A_1,\dots,A_k \in \mathfrak A} \mathbf{1}_{A_{1} \cap \dots \cap A_{k}}.\tag{1} $$ Though the formula (1) is very basic, but is very useful for computing factorial moments. Does it have a name? Or maybe there is a reference?