Arrange 5 red queens and 3 blue queens on a 5×5 board so that no queen attacks a queen of the other colour.
depends on what you mean by different way.
This is the only way known to me. You can of course set up different variations of this by turning the chess board (queens) by 90° or flipping all thequeens across.
I don't know about n*n chessboards.
with 5 red and 3 blues ones on the n*n board. well lets just say the bigger the chess board the more variations there is (with just 8 figures).
but you probably meant that the numbers of figures on the chess board is also bigger
I am not that familiar with probability. It'd take some time to notice the pattern on bigger chess boards and then relate it to the smaller ones with equation.