In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other? In how many different ways can we place $8$ identical rooks on a chess board so that no two of them attack each other?
I tried to draw diagrams onto a $8\times8$ square but I'm only getting $16$ ways.  Does that sound right?
Thanks for the help!

For two rooks to be attacking each other, they must either share a row or a column. For two rooks to not
attack each other they must not share a row and they must not share a column.
8 rows and 8 columns need to be chosen once each. A set of eight rows can
be chosen in $\binom{8}{8} = 1$ ways and a set of columns can be chosen in $\binom{8}{8} = 1$ ways.
Ordered pairs can be generated from the two sets in $8!$ ways.
Given the first coordinate of the first ordered pair (can be anything wlog),
there are $\binom{8}{1}$ ways to choose the second coordinate for the first ordered pair.
Given the first coordinate of
the next ordered pair, there are $\binom{7}{1}$ ways to choose the second coordinate.
Because $\binom{n}{1} = n$ the total number of ways to create the coordinates is $n(n−1)(n−2)...(1) = n!$.
In our case, $n = 8$ so there are $8!$ ways to generate valid coordinates or ways place the rooks.
In total, there are $\binom{8}{8} 8! = 8!$ ways to place the rooks.
