Why do we need this? What's wrong with good old "there are n ways to pick first element, n-1 ways to pick second, etc and then apply burnside to glue together things that we treat as identical"?
Why would anyone play this game when he could just diagonalize (nxm)x(nxm) matrix in F2 and get a basis? (in case of non-singular, otherwise impossible)
upd: misunderstood the game, it's about solving Ax=b in F2, so Gauss elimination.
Deleted my account there a year ago after it started sending me lots of questions of type "what are the first signs that you have illness X" where X varied from lyme disease to ebola. After finding that I match all the known diseases, I decided that this BS was not really worth an account there.
SO is so much better in terms of content quality - the amount of BS stupid questions on Quora is just hitting over the roof