I need some help understanding how to derive the moment generating function of a binomial distribution. I know that the solution is supposed to be:

$\displaystyle m(t) = (pe^t+q)^n$

so far I have:

$\displaystyle m(t) = \sum_{y=0}^{\infty} e^{ty}p(y) = \sum_{y=0}^{\infty} e^{ty} \binom{n}{y}p^y q^{n-y} = \sum_{y=0}^{\infty} \frac{n!}{y!(n-y)!} (pe^t)^y q^{n-y}$

at which point I don't know how to carry on.