Problem - 4263
In the following $(8\times 5)$ grid, how many shortest routes are there from point $A$ to point $B$?
A shortest route always required $8+5=13$ steps, among which $8$ needs to be go east. Therefore, the answer is $$C_{13}^{8} = C_{13}^5 = \frac{13!}{8!\cdot 5!}$$
In general, the number of shortest routes in a $(a\times b)$ grid is $$C_{a+b}^a = C_{a+b}^b=\frac{(a+b)!}{a!\cdot b!}$$