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
