Consider an \(8 \times 8\) chessboard. The number of ways 8 rooks can be placed on the board such that no two rooks are in the same row and no two are in the same column is

Rook 1 in row 1 can take any of 8 columns, rook 2 in row 2 any of the remaining 7, and so on, so the columns form a permutation of the 8 columns: 8! = 40,320 ways. The value 8 counts only the choice for one row, and 7! leaves out the 8 choices for the first rook.