Example 1: A and B participate in an election, A gets $m$ votes, B gets $n$ votes, $m>n$. Question: In the process of counting the $m+n$ votes one by one, how many possible vote counting records are there where A's vote count is always leading?
The voting record is represented by an $(m+n)$-tuple $\left(a_{1}, a_{2}, \cdots, a_{m+n}\right)$. When the $k$-th vote is for candidate A, we set $a_{k}=1$; if it is for candidate B, we set $a_{k}=-1$ (for $k=1,2, \cdots, m+n$). Let $b_{k}=a_{1}+a_{2}+\cdots+a_{k}$ (for $k=1,2, \cdots, m+n$). By connecting the following grid points from left to right: $\left(1, b_{1}\right),\left(2, b_{2}\right), \cdots,\left(k, b_{k}\right), \cdots$, $\left(m+n, b_{m+n}\right)$, we obtain a broken line with $m+n-1$ segments. Since candidate A's votes are always leading, we have $b_{k}>0$ (for $k=1, 2, \cdots, m+n$) and $b_{1}=1, b_{m+n}=m-n$. Thus, the broken line is a line connecting $(1,1)$ and $(m+n, m-n)$ that does not intersect the $x$-axis. Let the set of all such broken lines be denoted by $N$, and the set of all voting records that meet the conditions be denoted by $M$. Then $M$ and $N$ are in one-to-one correspondence. By the corollary, we have
$$
|N|=|M|=C_{m+n-1}^{m-1}-C_{m+n-1}^{m}=\frac{m-n}{m+n} C_{m+n}^{m} .
$$