Suppose I have a 5×5 grid of squares. I would like to fill in 15 checkmarks in the squares such that (1) each of the 25 square cells contains at most one checkmark, (2) each row has exactly 3 checkmarks, and (3) each column has exactly 3 checkmarks. How many ways are there to fill in these 15 checkmarks?

More generally, suppose I have an $ n \times n $ square grid, and I would like to fill in $ mn$ checkmarks such that (1) each of the $ n^2$ square cells contains at most one checkmark, (2) each row has exactly m checkmarks, and (3) each column has exactly m checkmarks. How many ways are there to do so? If $ m=1,$ I think the answer is $ n!$ . But I am not sure about the general case.

Also, if I have an additional restriction that no checkmarks on the diagonal, i.e., no checkmark in the (1,1), (2,2),… (n,n) cells. How many ways are there?

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