In how many ways can you place 8 rooks on a 8x8 chessboard so that no row or column contains more than one rook if

82 = 64 82 = 64 choices for that.

Now, for the second one, we can't be in the row or column of that first one, so leaving us with 72 = 49 72 = 49 choices.

Then so on, we have 62 = 36 62 = 36 for the third one, 2525 for the fourth one, and so on ...

But, however, we have to remember the rooks are not labeled, thus it doesn't matter specifically about a specific rook's position.

Thus, we have a total of (8!) 2 8! = 40320 (8!) 2 8! = 40320 ways.