In how many different ways can we place 8 identical rooks on a chess board so that no two of them attack each other?
Since rook in the $(i,j)$-th position can attack row $i$ and column $j$, for $i,j = 1,2,\dots,8$, it is equivalent to having the rooks on the diagonal. Since there are $8$ diagonal positions, the problem then becomes the number of ways of arranging the $8$ rooks, i.e. $8!$
You can only have one rook per column. Therefore for each column there is one rook. Call the rook in the first column $R_1$ in the second $R_2$ and so on. Therefore you must assign a row from $1$ to $8$ to each $R$ without repeating. There are $8!$ Ways to do so which is the same as $40320$.