Question 65
A railway passes through four towns $A$, $B$, $C$, and $D$. The railway forms a complete loop, as shown on the right, and trains go in both directions. Suppose that a trip between two adjacent towns costs one ticket. Using exactly eight tickets, how many distinct ways are there of traveling from town $A$ and ending at town $A$? (Note that passing through $A$ somewhere in the middle of the trip is allowed.)
