Question

The diagram shows three famous RSM cafes, R, S, and M. It takes one step to move from one of these cafes directly to another.

How many different ways are there to start at R and end at M in exactly ten steps?
One possible way is
R − M − R − S − M − S − R − M − S − R − M.

Answer 1

341

Explanation:

The path from R can end at M only if there are 0, 3, 6, or 9 clockwise steps together with 10, 7, 4, or 1 counterclockwise steps (repsectively).

In 10 steps, the total number of ways that can happen will be ...

10C0 +10C3 +10C6 +10C9 = 1 +120 +210 +10 = 341

where nCk = n!/(k!(n-k)!), the number of combinations of n things taken k at a time.

There are 341 different ways to start at R and end at M in exactly 10 steps.

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Additional comment

There are 1024 possible ways to take 10 steps. (A step can be only CW or CCW. 2^10 = 1024.) Of those, 341 end at M, 341 end at S, and the remaining 342 end at R.