Moment-generating function of a binomial distribution ($M_X(t)=(1-p+p{e}^t{)}^n$)

For a binomial distribution with parameters $n$ and $p$, the binomial theorem gives the displayed moment-generating function, finite for every real $t$. Products of these moment-generating functions show that the sum of independent binomial random variables with the same success probability again has a binomial distribution, with trial counts added.