Inspired by this question and some related ones here, I'd like to find the number of ways, $S(k, r, c)$, to place $k$ ones and $rc-k$ zeros in an $r\times c$ array so that the row sums and column sums are all even. Obviously, we'll require that $k\le rc$ and $k$ is even.
It seems reasonable to require that a solution that can be made from another by permuting the rows and the columns shouldn't be counted as distinct. For example, with $k=12, r=4, c=6$ we would have at least these inequivalent solutions: $ \left[\begin{array}{cccccc} 0 & 1 & 1 & 1 & 0 & 1\\ 1 & 0 & 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \end{array}\right] \qquad\text{and}\qquad \left[\begin{array}{cccccc} 1 & 1 & 1 & 0 & 1 & 0\\ 0 & 0 & 1 & 1 & 1 & 1\\ 1 & 1 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] $ if for no other reason than no row or column permutation can change the one on the left to have a row consisting of all zeros.