Example 3 As shown in Figure $30-1$, it is an $m \times n$ chess street, where horizontal lines represent horizontal streets, and vertical lines represent vertical streets. A person walks from $A$ to $B$ along the streets. How many shortest paths are there?
Solution: The person may first walk several segments of horizontal streets, several segments of vertical streets, then several segments of horizontal streets, several segments of vertical streets, ..., but the total number of segments of horizontal and vertical streets are $n$ and $m$, respectively. Let walking one segment of a horizontal street be denoted as $a$, and walking one segment of a vertical street be denoted as $b$. Then, each time the person completes the entire journey (by the shortest route), it corresponds to a permutation of $n$ $a$s and $m$ $b$s; conversely, each permutation of $n$ $a$s and $m$ $b$s also corresponds to a way of completing the entire journey, i.e., there is a 1-1 correspondence between the shortest route walk methods and the different permutations of $n$ $a$s and $m$ $b$s. The number of different permutations of $n$ $a$s and $m$ $b$s is $C_{m+n}^{n}$, so the number of shortest route walk methods from $A$ to $B$ on the chessboard street is $C_{m+n}^{n}$.