10.2. Find the number of different arrangements of 8 rooks on different white squares of an 8 by 8 chessboard such that no rook attacks another. A chess rook attacks all squares in the row and column it occupies.
Answer. $576=(4!)^{2}$.

Solution. From the fact that no rook attacks another, it follows that there is exactly one rook in each row and each column of the board. Note that the white cells of the columns with odd numbers (first type) are located in rows with even numbers, while the white cells of the columns with even numbers (second type) are located in rows with odd numbers. Therefore, rooks standing in columns with numbers of different parity cannot attack each other. Consequently, the total number of the desired arrangements is equal to the number of arrangements of 4 rooks in cells of the first type, which is 4!, multiplied by the number of arrangements of 4 rooks in cells of the second type, also 4!.

Criteria. Noted that rooks standing in columns with numbers of different parity cannot attack each other: 3 points. Correct calculation of the number of arrangements of 4 rooks in cells of each type: 2 points. Their multiplication: 2 points.