2018 AMC 10B Problem 5

Question

How many subsets of

\{2, 3, 4, 5, 6, 7, 8, 9\}

contain at least one prime number?

128

192

224

240

256

Count the subsets with no prime and subtract from the total number of subsets

The non-prime elements are

4, 6, 8, 9;

any subset of just those contains no prime

Count the complement. The set has

2^8 = 256

subsets total. A subset avoids every prime exactly when it sticks to the non-primes

\{4, 6, 8, 9\},

and there are

2^4 = 16

of those. So

256 - 16 = 240

subsets contain at least one prime. Thus, D is the correct answer.

Problem 5 in other years