Consider the following:
A bus driver pays all tolls, using only nickels and dimes, by throwing one coin at a time into the mechanical toll collector. Find a recurrence relation for the number of different ways the bus driver can pay a toll of n cents (where the order in which the coins are used matters).
The solution is $a_n = a_{n-5} + a_{n-10}$ With the base cases being $a_0 = 1$ and $a_5 = 1$. Could someone help clarify why this is the solution? I understand the second base case. $a_5 = 1$ because there is 1 way to pay a 5 cent toll. But I don't understand the first base case or the relation itself. What process can I follow to solve this word problem as well as others?