Example 7 To move a chess piece from point $(0,0)$ to point $(m, n), m, n \in \mathrm{N}$, and it is stipulated that the movement must be along the coordinate axes, and the direction of movement can only be changed at points where both coordinates are integers. How many shortest paths are there to move the chess piece?
Solution: According to the rule, the length of the route from point $(0,0)$ to point $(m, n)$ is $m+n$, and any shortest route consists of moving $m$ units in the positive direction of the $x$-axis and $n$ units in the positive direction of the $y$-axis.

Let moving one unit in the positive direction of the $x$-axis be denoted by $a$, and moving one unit in the positive direction of the $y$-axis be denoted by $b$. Then, any such shortest route can be represented by a sequence of $m$ $a$s and $n$ $b$s (thus, it can be viewed as a permutation problem with repetition of $a$ and $b$, repeated $m+n$ times). In fact, each such sequence contains $m+n$ letters, and we only need to decide which positions will be occupied by the letter $a$ (the rest will naturally be occupied by the letter $b$), thus the number of shortest routes is $C_{m+n}^{m}$.