Notice that switching one row with another in a table that satisfies the desired property gives another table that satisfies it (and the same goes for columns). If so we can always get by row switching to a state where every i-th row has i red squares and then column switch to a state where every i-th column has i squares. This process can be undone in (n!)^2 ways (those are the ways to assign an index from {1,2,...,n} to every row and also to every column) so we arrive to the conclusion that the result is exactly (n!)^2