Let $N$ be a number of fixed points in a random permutation of $\{1,\ldots,n\}$; each permutation has an equal probability. The problem I'm working on ask me to compute $\mathbb{E}[N]$, $\mathrm{Var}(N)$, and the factorial moment $\mathbb{E}[(N)_k]$ where $(N)_k$ is the falling factorial defined as $(N)_k:= \frac{N!}{(N-k)!}$ for $k=1,\ldots,N$.
I'm able to find $\mathbb{E}[N] = 1$ and $\mathrm{Var}(N) = n-\frac{1}{2}$ but have no idea how to proceed with $\mathbb{E}[(N)_k]$. I'd appreciate any suggestion on how I should approach this.