For given $n,k$, I am looking to simplify
$$f(i,j)=\sum_{\ell=1}^n \left(\begin{array}{c}\ell \\ i\end{array}\right)\left(\begin{array}{c}\ell \\ j\end{array}\right)\left(\begin{array}{c}n-\ell \\ k-i\end{array}\right)\left(\begin{array}{c}n-\ell \\ k-j\end{array}\right)$$ in terms of $i$ and $j$.
Using the Chu-Vandermonde identity, I got $$\sum_{i,j}f(i,j) = n\left(\begin{array}{c}n \\ k\end{array}\right)^2.$$.
I feel that some trick with generating functions might work, but I don't know how to move forward with it.
EDIT: Updated $\sum_{i,j} f(i,j)$ expression because the original statement was false.