A shortest route requires $5+4+3=12$ steps, among which $5$, $4$ and $3$ steps should follow the three directions, respectively. Therefore the answer is $$\frac{12!}{5!\cdot 4!\cdot 3!}$$

In general, the number of shortest route on a $(a\times b\times c)$ grid is $$\frac{(a+b+c)!}{a!\cdot b!\cdot c!}$$