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...") |
|||
| Line 1: | Line 1: | ||
| + | ==Problem== | ||
| + | |||
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> | ||
| + | |||
| + | ==Solution== | ||
| + | |||
| + | 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 
. 
