How many ways can you place n rooks on a chessboard?

For 8 rooks

Without conflict (i. e. there is no row or column can contains 2 rooks)

then there are 8 choices for the first one in the first column, 7 choices in the second, ... 1 choice in the last (eighth) column for a total of

8!=40320 ways.

Unlimited conflicts

there are 64 choices for the first, 63 for the second, ... 57 for the last, for a total of 64!/56!=178462987637760 ways

Similarly, for n rooks, with unlimited conflicts, there are

64! / (64-n) ! ways