Rooks or castles are chess pieces that are only allowed to m

Rooks (or castles) are chess pieces that are only allowed to move horizontally or vertically on a chessboard. The following question on rook placements on a standard chessboard assumes that rooks must be placed on one of the 64 allowed positions on the board, and no two rooks can share the same position. How many different ways are there of placing 8 indistinguishable rooks on a standard 8 × 8 chessboard

(a) if there are no restrictions on their placement, other than those stated above?

(b) if no rook can be in a position that attacks another? [A rook can attack another if and only if they are on the same row or column of the chessboard.]

Solution

(a) if there are no restrictions on their placement, other than those stated above?

The answer is 64!/(56!)(8!). My reasoning for this is that there are 64! possibilities and then you have to divide out by the 56 empty squares as well as the 8 squares the rooks are placed on

b)

the answer is 64! / (8!) because no rook can be in a position that attacks another