15. A and B participate in an election, A gets $m$ votes, B gets $n$ votes, $m >$ $n$, ask: In the process of counting $m+n$ votes one by one, how many vote counting records are there where A's vote count is always leading?
15. The starting state of the vote counting corresponds to the origin $O$, if A gets 1 vote, then a diagonal of a unit square with a slope of 1 is drawn to the right; if B gets 1 vote, then a diagonal of a unit square with a slope of -1 is drawn to the right, the endpoint of the previous segment is the starting point of the next segment, and the endpoint of the entire broken line is $P(m+n, m-n)$. Each vote counting process where A remains in the lead corresponds to a broken line connecting $O$ and $P$, and except for $O$, it has no common points with the x-axis. Since the first segment of this broken line must be the line connecting $O$ and $Q(1,1)$, the broken line connecting $O$ and $P$ and having no common points with the x-axis except for $O$ corresponds to the broken line connecting $Q$ and $P$ and having no common points with the x-axis. Therefore, the set of all vote counting processes where A's votes always remain in the lead and the set of broken lines connecting $Q$ and $P$ and having no common points with the x-axis form a 1-1 mapping, so the number of all such processes is $\frac{m-n}{m+n} C_{m+n}^{m}$.