Suppose two candidates A and B poll the same number of votes, $n$ each, in an election. The counting of votes is usually done in some arbitrary order and therefore, during the counting process A may lead for some period and B may lead for some period. Prove that the number of ways of counting the votes such that A never trails B is the $n^{th}$ catalan number.
I know that in such questions we are supposed to analyse the problem by dropping a few terms. For example, in this one, we can drop the $2n^{th}$ vote and check it for $2n-1$ votes, but it doesn't seem to be correct since I won't get catalan number. Can someone please help in getting a combinatorial argument/proof for this? Thanks!