2004 AMC 12B Problems/Problem 20
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Problem
Each face of a cube is painted either red or blue, each with probability 
. 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?
Solution
There are 
possible colorings of the cube. Consider the color that appears with greater frequency. The property obviously holds true if 
or 
of the faces are colored the same, which for each color can happen in 
ways. If 
of the faces are colored the same, there are 
possible cubes (corresponding to the possible ways to pick pairs of opposite faces for the other color). If of the faces are colored the same, the property obviously cannot be satisfied. Thus, there are a total of 
ways for this to occur, and the desired probability is 
.
See also
| 2004 AMC 12B (Problems • Resources) | ||
| Preceded by Problem 19 | Followed by Problem 21 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
