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Why would a bug be moving along the edges of a cube?

Source: AMC 12 2006A, Problem 20

February 5, 2006
geometry3D geometryprobabilityAMC

Problem Statement

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? <spanclass=′latex−bold′>(A)</span>12187<spanclass=′latex−bold′>(B)</span>1729<spanclass=′latex−bold′>(C)</span>2243<spanclass=′latex−bold′>(D)</span>181<spanclass=′latex−bold′>(E)</span>5243 <span class='latex-bold'>(A) </span> \frac {1}{2187} \qquad <span class='latex-bold'>(B) </span> \frac {1}{729} \qquad <span class='latex-bold'>(C) </span> \frac {2}{243} \qquad <span class='latex-bold'>(D) </span> \frac {1}{81} \qquad <span class='latex-bold'>(E) </span> \frac {5}{243}
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