1. The problem statement, all variables and given/known data There are 7 different routes from A to B, 4 different routes from B to C, and two different routes from C to A. What are the possible routes from A to C and back, allowing any route to be traversed once in each direction. 2. Relevant equations 3. The attempt at a solution So I think I have it, but I just need some verification since we've only had one lecture and my high school combinatoric lessons (if you could call them that) weren't very "in depth" and several years ago. So I figure these are the possibilities: A to B to C to A gives 7*4*2 = 56 possible routes A to B to C to B to A gives 7*4*4*7 = 784 possible routes A to C to A gives 2*2 = 4 possible routes Altogether that makes 844 possibilities. However, I'm wondering if this one "counts" as well: A to B to A to C to A gives 7*4*7*2*2 = 784, however I'm not sure if that double counts some of the routes from above. Any help would be greatly appreciated. Edit: I also just thought of A to C to B to A, but that one I think double counts. Thanks again.