2024 AMC 10A Problem 24

ProbabilitySolid GeometryHard

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?

\text{(A)}\;\frac{1}{54}

\text{(B)}\;\frac{7}{54}

\text{(C)}\;\frac{1}{6}

\text{(D)}\;\frac{5}{18}

\text{(E)}\;\frac{2}{5}