Final answer:
To find the number of ways to color a 6x6 grid with 2 red squares in each row and column, we consider the combinations for placing red squares in one column, then raise this to the sixth power to account for all columns, similarly considering rows.
Step-by-step explanation:
To solve the problem of coloring a 6x6 grid with red and blue squares, where each row and column contains exactly two red squares, we can use combinatorial methods. We are essentially placing two red squares in each row and each column without violating the given condition.
We can imagine fixing two red squares in the first column, there are 15 combinations (since we choose 2 positions out of 6 for the red squares). For each of these, the red squares in the second column can again be placed in 15 ways, but we must avoid the positions taken by red squares in the first column. This pattern continues for each subsequent column, which yields the formula 15^6.
However, this does not take into account the permutational aspect of the rows, so we must also consider the number of ways to arrange 2 red squares in each row. The number we calculated before assumes a fixed arrangement of reds in the columns, but since each row is identical, we can also calculate the number of ways to arrange red squares in one row, then raise this to the sixth power. The principle remains similar to the column approach.