【Example 4】Let $n \geqslant 3$ be an integer. Try to find the number of paths starting at $(0,0)$, ending at $(2 n, 0)$, that lie above the $x$-axis and do not intersect the $x$-axis.
Observing Figure 20, it is easy to see that the number of such paths is the same as the number of paths starting at $(1,1)$ and ending at $(2 n-1,1)$, which are above the line $y=1$ and intersect it at most at one point. If the origin of the coordinate system is translated to the point $(1,1)$, it immediately transforms into a path starting at $(0,0)$ and ending at $(2 n-2,0)$, which is above the $x$-axis and does not intersect it. Therefore, the number of such paths is $C_{2 n-2}^{n-1} / n$.