Summations of Products of Binomial Coefficients

Question

It seems that the following equation is true for any $k<N$ (after verifying a lot of $k$ and $N$):

\begin{equation} \sum_{j=k}^{N} (-1)^j \begin{pmatrix} N\\j \end{pmatrix} \begin{pmatrix} j\\k \end{pmatrix} = 0 \end{equation}

But I want to proof it, any ideas?

Answer 1

Hint: call this number $a_k$ and compute the polynomial $\sum_{k=0}^{N} a_k x^k$.