Recursion, Catalan Numbers & Generating Functions · Challenge problem

Tickets cost $\$5$. Twelve people wait in line: seven will pay with a $\$5$ bill and five with a $\$10$ bill. The cashier starts with no money and gives each $\$10$ payer a $\$5$ bill in change. If all $\binom{12}{5}$ orders of the kinds of bills are equally likely, what is the probability that the cashier is never unable to make change? Express your answer as a common fraction.