The theorem that the $s$th factorial moment for the sum of $N$ events is $s!$ times the sum of the expectations for any $s$ of the events occurring simultaneously has been proved by induction. The applications of this result in obtaining easily the moments of a number of distributions arising from a sequence of observations belonging to two continuous populations and other cases have been demonstrated.
"A Theorem on Factorial Moments and Its Applications." Ann. Math. Statist. 29 (1) 254 - 261, March, 1958. https://doi.org/10.1214/aoms/1177706723