Question

There are three roads between towns A and B. There are 6 roads between towns B and C.
How many different routes may one travel from town A to town C through town B?

Answer 1

It seems to me that there would be 72 ways.

There are 6 unique ways to get from town A to town B and back. If the roads are 1 to 3, the combinations are: 1–2, 1–3, 2–1, 2–3, 3–1, and 3–2.

There are 12 unique ways to get from B to C and back. If the roads are 4 to 7, the combinations are: 4–5, 4–6, 4–7, 5–4, 5–6, 5–7, 6–4, 6–5, 6–7, 7–4, 7–5, and 7–6.

A person can take road 1 from A to B, and then have 12 ways to get from B to C and back before taking road 2 back to A. That means that for every unique A to B to A combination there are 12 unique B to C to B combinations. That should be 6 X 12, or 72.