2021 Fall AMC 10A Problems/Problem 13
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Problem
Each of balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other
balls?
Solution
Note that for this restriction to be true, there must be balls of each color. There are a total of
ways to color the balls, and there are
ways for three balls chosen to be painted white. Thus, the answer is
.