2.3 ** A $4 \times 4$ chessboard consisting of 16 squares, 8 of which are to be colored red, such that each row and each column contains exactly 2 red squares. How many ways are there to color the squares?
Parse the first line: coloring 2 red cells, there are $\mathrm{C}_{4}^{2}=6$ ways.
After coloring the first line, there are three scenarios:
(1) The red cells in the second line are in the same columns as those in the first line, in which case the coloring of the remaining lines is uniquely determined.
(2) The red cells in the second line are in different columns from those in the first line, in which case, the third line has $\mathrm{C}_{4}^{2}=6$ ways of coloring, and the coloring of the fourth line is then determined.
(3) The red cells in the second line have exactly one in the same column as those in the first line, and there are 4 such ways. In this case, the red cells in the third line must have one in a different column from the previously colored red cells, so the third line has 2 ways of coloring, and the coloring of the fourth line is then determined.
Therefore, there are a total of $6 \cdot(1+6+4 \times 2)=90$ different ways of coloring.