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

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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 $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?

$\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$

Solution

Consider finding the number of subsets that do not contain any primes. There are four primes in the set: $2$, $3$, $5$, and $7$. This means that the number of subsets without any primes is the number of subsets of $\{4, 6, 8, 9\}$, which is just $2^4 = 16$. 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 $2^8 = 256$. Thus, the answer is $256 - 16 = 240$. $\boxed{D}$

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