Example 1-30 Many streets are built in a grid pattern. Starting from home to the destination, one needs to walk $m$ streets to the east and $n$ streets to the north. How many shortest paths are there from home to the workplace?
If we consider home as the $(0,0)$ point and the workplace as the $(m, n)$ point, the problem can be transformed into finding how many shortest paths there are from the $(0,0)$ point to the $(m, n)$ point (Figure 1-13).

The path from the $(0,0)$ point to the $(m, n)$ point must involve crossing $m$ streets in the $x$-axis direction and $n$ streets in the $y$-axis direction. That is, each path corresponds to a permutation of $m$ $x$s and $n$ $y$s, forming a total of $m+n$ steps. This can be viewed as selecting $m$ cells out of $m+n$ cells to fill with $x$, and the remaining $n$ cells to fill with $y$. The number of such permutations is $C(m+n, m)$.