A common problem in Combinatorial Analysis Please help me prove the fact below:
$$
\sum_{n=1}^N \frac{N!}{n!(N-n)!}\frac{(-1)^{n-1}n}{n+x} = \frac{N!}{\prod_{n=1}^N (n+x)}.
$$
I think this problem is common, but it is really hard for me to solve.

It is a straightforward consequence of the residue theorem. The rational function:
$$ f(x) = \frac{1}{\prod_{n=1}^{N}(x+n)} $$
has simple poles in $x=-1,-2,\ldots,-N$, hence:
$$ f(x) = \sum_{j=1}^{N} \frac{c_j}{x+j} $$
where:
$$ c_j = \operatorname{Res}\left(f(z),z=-j\right).$$
