One Bill in the Till

A ticket costs $5. Twelve customers line up in uniformly random order: six of them pay with a $5 bill and six pay with a $10 bill (and need $5 change). The cashier starts with exactly one $5 bill in the till and no other money, and she can only give change from bills she has received or the one she started with. What is the probability that she can give every customer correct change when they reach the front? Give a reduced fraction.

• counting paths
• distributions

Source: Variant of the classical theatre-queue (ticket-change) problem. Statement written for AxiomIQ.