3.1 Q5

There are three different routes connecting city A to city B. How many ways can a round trip be made from A to B and back? How many ways if it is desired to take a different route on the way back?

Let the different routes between A & B be defined as x, y, z. Assuming that route x-y is a different route from y-x, then there are 9 possible round trips between A & B (x-x, x-y, x-z, y-y, y-x, y-z, z-z, z-x, z-y), and 6 using a different route on the way back (x-y, x-z, y-x, y-z, z-x, z-y).

If x-y and y-x are judged to be the same route, however, then there are 6 possible round trips between A & B (x-x, x-y, x-z, y-y, y-z, z-z), and 3 different routes (x-y, x-z, y-z).