6. As shown in Figure 3, fill in the "nine-square grid" with 4 zeros, 4 ones, and 1 three. The number of ways to fill it such that the sum of the numbers in each row and each column is odd is $(\quad)$.
(A) 45
(B) 60
(C) 90
(D) 120
6. A.

Since the sum of each row (column) is odd, each row (column) must contain an odd number of odd numbers, and there are only five odd numbers. Therefore, one row (column) must contain three odd numbers, and the other two rows (columns) must contain one odd number each. Thus, the five odd numbers must be distributed in the 3x3 grid in a vertical or horizontal alignment, as in one of the nine scenarios: the five odd numbers are located in the first row and the first column (as shown in Figure 6). In each scenario, since the odd number 3 can be placed in any of the five positions, the total number of ways that meet the requirement is 45.