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Thanks

Rishi

Here, the important observation to make is about the faces -- every step must make progress from P to Q, which means that the path can only traverse two faces. If so, you can "unfold" the two faces and get something like this:

P_ _ _ _ |_|_|_|_| |_|_|_|_| Q

Here, the number of paths is easy to count -- six steps are required, two of them must be down (and the other four are to the right), so the total number of possibilities is 6 choose 2 = 15.

However, you can start with any face that includes P (3 choices) and from that face, go to any of the two faces it touches that include Q (2 choices).

The total number, then, is 2 × 3 × 15 = 90.

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