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2018 AMC 12B Problem 5

Problem 5 of 25EasierCounting & Probability

How many subsets of {2,3,4,5,6,7,8,9}\{2, 3, 4, 5, 6, 7, 8, 9\} contain at least one prime number?

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Solution

The set has 88 elements, giving 28=2562^8=256 subsets. The subsets with no prime use only the four non-primes {4,6,8,9},\{4,6,8,9\}, and there are 24=162^4=16 of these. So the number containing at least one prime is 25616=240.256-16=240. Thus, the correct answer is D.

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Concepts: complementary counting · subsets

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.