Example 13: In a table tennis match between A and B, A finally wins with a score of $21: 16$. Find the number of score sequences in which A has been leading throughout the match.
Solution: The initial state of the game corresponds to the origin $O$, a score of 1 point by A connects a diagonal of a unit square with a slope of 1, and a score of 1 point by B connects a diagonal of a unit square with a slope of -1. According to the problem, the endpoint of this broken line should be $P(37,5)$. Therefore, the score sequence where A is always leading corresponds to connecting $O$ and $P$, and has no common points with the $x$-axis (except for $(0,0)$), and the first segment can only be the broken line from $(0,0)$ to $Q(1,1)$.

Thus, in fact, we only need to consider the number of broken lines connecting $(1,1)$ and $(37,5)$, and having no common points with the $x$-axis, and such broken lines have $C_{36}^{20}-C_{36}^{21}$. Therefore, the number of score sequences that meet the requirements is $C_{36}^{20}-C_{36}^{21}$.