How many ways can we place $8$ rooks on an $8\times 8$ chessboard, if $4$ are plastic and $4$ are glass? Assuming they don't attack their own type (can share rows and/or columns).
I'm also curious, if they couldn't share rows/columns with their own type, would this be the same as $8$ identical rooks? Which is $8! = 40,320$? I assume not, but why?
For more details on rook polynomials see my answer here and you can also check out this rook polynomial calculator.
Firstly there are $\binom{8}{4}^24!$ placements of the plastic rooks since we choose 4 rows from 8 for the rooks in $\binom{8}{4}$ ways then order the rooks in 4 of the 8 columns in $\binom{8}{4}4!$.
Then, note that each placement removes four $1\times 1$ disjunct squares from our board.
$$\begin{array}{c|c|c|c|c|c|c|} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline 1 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\phantom{H}} &\bbox[white,10px]{\Large\unicode{x265c}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 2 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 3 &\bbox[white,10px]{\Large\unicode{x265c}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 4 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 5 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 6 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\unicode{x265c}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 7 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\unicode{x265c}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 8 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline\end{array}$$
$$\downarrow\text{plastic rooks become forbidden squares}\downarrow$$
$$\begin{array}{c|c|c|c|c|c|c|} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline 1 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\phantom{H}} &\bbox[silver,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 2 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 3 &\bbox[silver,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 4 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 5 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 6 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[silver,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 7 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[silver,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 8 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline\end{array}$$
$$\downarrow\text{rows and columns can be rearranged to recontres-type}\downarrow$$
$$\begin{array}{c|c|c|c|c|c|c|} & 4 & 1 & 3 & 7 & 2 & 5 & 6 & 8 \\ \hline 1 &\bbox[silver,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 3 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[silver,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 6 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[silver,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 7 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[silver,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 2 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 4 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 5 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 8 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline\end{array}$$
$$\downarrow\text{glass rooks placed on recontres-type board}\downarrow$$
$$\begin{array}{c|c|c|c|c|c|c|} & 4 & 1 & 3 & 7 & 2 & 5 & 6 & 8 \\ \hline 1 &\bbox[silver,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\unicode{x265c}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 3 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[silver,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 6 &\bbox[white,10px]{\Large\unicode{x265c}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[silver,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 7 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[silver,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 2 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 4 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\unicode{x265c}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 5 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline 8 &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\phantom{H}} &\bbox[white,10px]{\Large\unicode{x265c}} &\bbox[white,10px]{\Large\phantom{H}} \\ \hline\end{array}$$
Therefore for each placement of the plastic rooks the glass rooks must be placed on a recontres-type chessboard. The rook polynomial for the forbidden subboard of such a board is given by
$$(1+x)^4$$
If we now call the rook polynomial for the $n\times n$ square board $R_n(x)$ then
$$R_n(x)=\sum_{k=0}^{n}\binom{n}{k}^2k!x^k$$
The rook polynomial of the complement to the forbidden subboard board $R_c(x)$ is given in rook theory by first expanding $(1+x)^4$
$$(1+x)^4=1+4x+6x^2+4x^3+x^4$$
Then replacing $x^k$ with $(-x)^kR_{8-k}(x)$ so that
$$R_c(x) = R_8(x) - 4xR_7(x) + 6x^2R_6(x) - 4x^3R_5(x) + x^4R_4(x)$$
the desired coefficient is $x^4$ since there are $4$ glass rooks to place on this complement board hence
$$\begin{align}[x^4]R_c(x) &= [x^4]R_8(x) - 4[x^3]R_7(x) + 6[x^2]R_6(x) - 4[x^1]R_5(x) + [x^0]R_4(x) \\&=\binom{8}{4}^24!-4\binom{7}{3}^23!+6\binom{6}{2}^22!-4\binom{5}{1}^21!+\binom{4}{0}^20!\end{align}$$
Since this is the placement count for $4$ glass rooks for each placement of plastic rooks we multiply
$$\begin{align}\text{desired count}&=\left(\binom{8}{4}^24!-4\binom{7}{3}^23!+6\binom{6}{2}^22!-4\binom{5}{1}^21!+\binom{4}{0}^20!\right)\cdot \binom{8}{4}^24!\\&=10\,678\,197\,600\end{align}$$
