Exercise 1. In how many ways can 7 rooks be placed on a $7 \times 7$ chessboard such that no rook can attack another?

A rook can attack another rook if it is on the same row or the same column.
Solution to Exercise 1

If we place 7 rooks such that no two rooks are in the same column, knowing that there are 7 columns, then there will be exactly one rook on each column (and similarly on each row).

There are 7 possible positions to place a rook on the first column. There are then 6 possible positions to place a rook on the second column without it being on the same row as the first rook. And so on, on the $\mathrm{k}$-th column there are 8 - $\mathrm{k}$ ways to place a rook without it being on the same row as a rook already placed.

Thus, there are $7!=7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1=5040$ ways to arrange the rooks in this manner.

If we denote $\sigma(\mathfrak{i})$ as the row number of the rook present on the $i$-th column in one of the configurations satisfying the statement, then $\sigma$ is what is called a permutation.

Grader's Comment

The problem was very well solved.