# Difference between revisions of "2018 AMC 10B Problems/Problem 5"

(Created page with "How many subsets of <math>\{2,3,4,5,6,7,8,9\}</math> contain at least one prime number? <math>\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \te...") |
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+ | ==Problem== | ||

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

<math>\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}</math> | <math>\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}</math> | ||

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+ | ==Solution== | ||

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+ | Consider finding the number of subsets that do not contain any primes. There are four primes in the set: <math>2</math>, <math>3</math>, <math>5</math>, and <math>7</math>. This means that the number of subsets without any primes is the number of subsets of <math>\{4, 6, 8, 9\}</math>, which is just <math>2^4 = 16</math>. The number of subsets with at least one prime is the number of subsets minus the number of subsets without any primes. The number of subsets is <math>2^8 = 256</math>. Thus, the answer is <math>256 - 16 = 240</math>. <math>\boxed{D}</math> |

## Revision as of 14:56, 16 February 2018

## Problem

How many subsets of contain at least one prime number?

## Solution

Consider finding the number of subsets that do not contain any primes. There are four primes in the set: , , , and . This means that the number of subsets without any primes is the number of subsets of , which is just . The number of subsets with at least one prime is the number of subsets minus the number of subsets without any primes. The number of subsets is . Thus, the answer is .