Ticket line

At a theater ticket office, there are $2n$ people waiting in line, in uniformly random order, to buy tickets. $n$ of them have only $5$-dollar bills, while the other $n$ people have only $10$-dollar bills. Unfortunately, the ticket seller has no change to start with. Each person wants to buy one ticket costing $5$ dollars. What is the probability that everyone can successfully buy their ticket without anyone having to change places, if there are $12$ people in line ($n = 6$)?