, 2014

Switching the definition of selection and non-selection, we see that an equivalent formulation is to select eight squares, with one in each row and in each column, and no black squares selected. We assume that the non-black squares are white.

Orient the chessboard so that the square in row $1$, column $1$ is white. Let $a_i$ be the column of the selected white square in row $i$, so $i \mapsto a_i$ is a permutation of $\{1, \dots, 8\}$ and $a_i$ has the same parity as $i$. There are $4$ choices for $a_1$ ($a_1 \in \{1, 3, 5, 7\}$), then $3$ for $a_3$ ($a_3 \in \{1, 3, 5, 7\} \setminus \{a_1\}$), $2$ for $a_5$, and $1$ for $a_7$. After these selections, we again have $4$ choices for $a_2$, $3$ for $a_4$, etc. All of these sets of choices are independent of each other, so in all we have $(4!)^2 = 576$ ways of performing our selection.