I'm stuck on part c) of this question. The answer key gives 182. I already know there are 14 ways to make the trip from city A and city B and vice versa. It appears 182 came from $14 \times 13$ but I don't get where 13 came from? If neither $R_8$ or $R_9$ were used on the trip there, one of the roads in the middle would need to be removed from the return trip so that would become $2 \times 4$ or $3 \times 3$.
By the way, what's the policy for this site regarding typing out questions instead of putting images for them? Normally I would try but if a person can't view images then they wouldn't be able to see the graph and wouldn't be able to help anyways.
Best Answer
To plan a round trip from $A$ to $C$ and back, where the return route is not the reverse of the first part from $A$ to $C$, we proceed as follows. Since there are $14$ ways to go from $A$ to $C$ we must choose one of those ways. Once that has been done, exactly one of the $14$ routes has been ruled out for reversal to get back to $A$ from $C$, leaving us $13$ choices for the return trip. This gives $14\cdot 13=182$ possible round trips.
Note that we do not have to consider the specific types of these trips in terms of how they go through $B$ (or avoid $B$). One of the trips for the first leg of the journey from $A$ to $C$ being chosen, it is the only one excluded on the way back.