10. Fill the nine blanks in Figure 2 with the numbers $1,2, \cdots, 9$, requiring that each row from left to right and each column from top to bottom increase sequentially. When 3 and 4 are fixed in the positions shown in the figure, the number of ways to fill the blanks is . $\qquad$
10.6.

The first two cells from left to right in the first row can only be arranged with $1, 2$, and the cell at the bottom right corner can only be 9. This leaves us to choose any two numbers from the remaining four numbers $5, 6, 7, 8$ to arrange in the top two cells of the right column (in ascending order), and the remaining two numbers are arranged in the first two cells of the bottom row (in ascending order). Therefore, the total number of methods is the number of ways to choose two numbers from the four numbers $5, 6, 7, 8$, which is $\mathrm{C}_{4}^{2}=6$ ways.