Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 5 / b / iii / Solution

Substitute $X$ into the polynomial identity and apply linearity of expectation to its finite sum. The factorial moments of the Poisson distribution in part (a)(iv) are all one, givingThus its $n$th moment counts set partitions: it equals the $n$th Bell number.