Question

Each face of a cube is painted either red or blue, each with probability $\tfrac12.$ The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?

$\dfrac{1}{4}$

$\dfrac{5}{16}$

$\dfrac{3}{8}$

$\dfrac{7}{16}$

$\dfrac{1}{2}$

Solution:

Fixing the orientation, there are $2^6 = 64$ colorings.

A coloring works if all six faces match ( $2$ ways), exactly five match ( $\binom{6}{5} \cdot 2 = 12$ ways), or four faces share a color with the remaining pair being opposite faces of the other color ( $3$ opposite pairs, $2$ colors, giving $6$ ways).

The total is $2 + 12 + 6 = 20,$ so the probability is $\dfrac{20}{64} = \dfrac{5}{16}.$

Thus, the correct answer is B.

2004 AMC 10B Problem 23

Solution:

Problem 23 in Other Years