So I want to find the number of ways to fill an $m\times n$ matrix with only 0s and 1s such that each row and column has an even number of 1s. I'm pretty stumped here. I've set up m+n equations summing the respective rows and columns yielding 0 mod2, but other than that I'm not too sure of a clean way to approach this.
The subsequent part asks for the same thing except that there's an odd number of 1s. Any hints and/or tips very much welcome.