Obviously there must be a rook in every row and every column. Starting
with the top row, it is clear that we may put our first rook on any one
of eight different squares. Wherever it is placed, we have the option of
seven squares for the second rook in the second row. Then we have six
squares from which to select the third row, five in the fourth, and so
on. Therefore the number of our different ways must be 8 × 7 × 6 × 5 × 4
× 3 × 2 × 1 = 40,320 (that is 8!), which is the correct answer.
How many ways there are if mere reversals and reflections are not
counted as different has not yet been determined; it is a difficult
problem. But this point, on a smaller square, is considered in the next
puzzle.