How many binary $N \times N$ matrices exist with a given row and column sum Given a $N \times N$ binary matrix (aka {0,1}- matrix) and an integer $k, k<N$, how many matrices can be constructed such that each row and column in the constructed matrix sum up to exactly k?
 A: There is no simple formula except for very small $k$ or $N-k$.  The most general asymptotic formula, though it seems to have not appeared in print yet, is by Liebenau and Wormald and the references therein.
A: It seems like I've solved this or seen it before.  The k=1 case is equivalent to the number of maximum independent sets  on the nxn Rooks graph.  The answer for k=1 is N!.
You're basically asking for how many ways to place nk rooks on a nxn board so that each row and column has exactly k rooks in it.  I think this problem has been solved before.  I'm on a cell phone so I don't have a link to the solution.  However, I'm very confident it has been solved before.
Update:  I take this back.  I don't think it has a known solution, although I've played around with the question myself.  This question states a solution isn't known for $N$ even and $k=\frac{N}{2}$:
Counting 2m X 2m 0-1 matrices with m ones in each row and each column.
The question is equivalent to your problem for a specific case of $k$.
A: Miller and Harrison (2013, link) solved this problem, even for arbitrary row and column margins per row and column ( link full text).
I think there was even a package, but sadly can't find it right now. Hope this helps.
