9. As shown in Figure 3, in a $10 \times 5$ grid, an ant starts from point $A$ and can only move to the right or upwards along the grid lines, but it is not allowed to pass through point $C$. The number of different paths the ant can take from point $A$ to point $B$ is $\qquad$.
9.1827 .

First, give a property: The number of the shortest paths from point $A(0,0)$ to point $X(m, n)$ along the grid lines is the combination number $\mathrm{C}_{m+n}^{n}$

Therefore, the number of paths from point $A(0,0)$ to point $B(10,5)$ is $\mathrm{C}_{15}^{5}$. Also, the number of paths from point $A(0,0)$ to point $C(5,3)$ is $\mathrm{C}_{8}^{3}$, and the number of paths from point $C(5,3)$ to point $B(10,5)$ is $\mathrm{C}_{7}^{2}$, so the number of the shortest paths from point $A$ to point $B$ that do not pass through point $C$ is $\mathrm{C}_{15}^{5}-\mathrm{C}_{8}^{3} \mathrm{C}_{7}^{2}=1827$.