Refer to an election in which two candidates Wright and Upshaw ran for dogcatcher. After each vote was tabulated, Wright was never behind Upshaw. This problem is known as the ballot problem.
Suppose that each candidate received exactly $r$ votes. Show that the number of ways the votes could be counted is $C_r$, the $r-$th Catalan number.
My issue with this is, what's the relation between Combinations and the Catalan number? I do have the answer, but I don't understand the logic behind the counting argument.