Question

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?

$\dfrac{1}{54}$

$\dfrac{7}{54}$

$\dfrac{1}{6}$

$\dfrac{5}{18}$

$\dfrac{2}{5}$

Answer: B

Concepts:basic probability cube geometry casework

Difficulty rating: 2380

Solution:

Every roll moves the bee one unit along $\pm x,\pm y,$ or $\pm z,$ so there are $6^4=1296$ equally likely sequences. There are two types of valid paths. A path around one square face has $3$ choices of coordinate plane, $4$ choices for the signs of its two axes, and $2$ choices for which axis is used first, giving $24.$ Otherwise all three coordinate directions are used, with one repeated: choose that axis in $3$ ways, choose its two nonadjacent positions in $3$ ways, order the other two axes in $2$ ways, and choose their three initial signs in $2^3=8$ ways. (The second step on the repeated axis must have the opposite sign.) This gives $3\cdot3\cdot2\cdot8=144.$ Hence there are $24+144=168$ favorable sequences, and the probability is $168/1296=7/54.$ Therefore, the answer is B.

2024 AMC 10A Problem 24

Answer: B

Difficulty rating: 2380

Solution:

Problem 24 in Other Years