Ticket Line

The Problem

A queue has $n$ people, each holding either a $5 bill or a $10 bill. Tickets cost $5. The cashier starts with no change. People are served in queue order; each person tenders their bill and requires correct change if it's a $10.

How many of the $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$ possible orderings of $n$ $5-holders and $n$ $10-holders will allow the cashier to serve everyone without running out of change?

(This is the ballot problem / Catalan number setup, classic in quant interviews.)