Question 159
A bee is moving in three-dimensional space. A fair six-sided die with faces labeled $A^+$, $A^-$, $B^+$, $B^-$, $C^+$, and $C^-$ is rolled. Suppose the bee occupies the point $(a, b, c)$. If the die shows $A^+$, then the bee moves to the point $(a+1, b, c)$, and if the die shows $A^-$, then the bee moves to the point $(a-1, b, c)$. Analogous moves are made with the other four outcomes. Suppose the bee starts at the point $(0, 0, 0)$ and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?
(A) $\frac{1}{54}$ (B) $\frac{7}{54}$ (C) $\frac{1}{6}$ (D) $\frac{5}{18}$ (E) $\frac{2}{5}$
(A) $\frac{1}{54}$ (B) $\frac{7}{54}$ (C) $\frac{1}{6}$ (D) $\frac{5}{18}$ (E) $\frac{2}{5}$