Expectation of $ X(X - 1) \ldots (X - k + 1) $, where $ X $ has a Poisson distribution. I am trying to calculate
$$
\text{E}[X(X - 1) \ldots (X - k + 1)],
$$
where $ \text{E} $ denotes the expectation operator and $ k \in \mathbb{N} $ is fixed.
I think I have to use the fact that the expectation of a sum of random variables is the sum of the expectations, but how can we apply it to this product?

Alternatively, the probability generating function $G$ for a Poisson random variable with mean $\lambda$ is $G(s)=E(s^X)=\exp(-\lambda(1-s))$. Differentiate $k$ times and set $s=1$ to get the answer. 
