Solution
= Solution
The <probability generating function> is
$$
G_X(s)=\sum_{m=1}^{\infty}\mathbb P(X=m)s^m.
$$
Differentiating $n$ times,
$$
G_X^{(n)}(s)
=\sum_{m=n}^{\infty}
\frac{m!}{(m-n)!}\mathbb P(X=m)s^{m-n}.
$$
At $s=0$, only the term $m=n$ remains. Hence
$$
\boxed{\mathbb P(X=n)=\frac{G_X^{(n)}(0)}{n!}}.
$$