In how many ways can you place 2 identical rooks on an 8 tim

In how many ways can you place 2 identical rooks on an 8 times 8 chessboard such that they will not be able to capture each other (i.e., they do not share the same row or column).

Solution

A rook on an 8x8 chessboard. if on the same row or same column will get captured.

2 rooks should be placed in 64 squares. So, the sample space is: ⁶⁴C₂ = 64! / (62! * 2!) = 64 * 63 / 2 = 2016.

Assume the first rook is placed in position: (1, 1), then the other rook which can be placed without being attacked is in 49 squares. And if the first rook is placed in position (1, 2), then the other rook can be placed without being attacked is in 49 more ways, And going this way, every time a rook is placed in one of the 64 squares, the other rook safely can be placed in 49 squares.

Therefore, the number of ways to place 2 identical rooks on an 8x8 chessboard such that they will not be able to capture each other is: 64 * 49 / 2 = 3136/ 2 = 1568.