How many subsets of {1,2,3,4,5,6,7,8} contain at least one p

[Max@Math Revolution]

[Math Revolution GMAT math practice question]

How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime number?

A. 60
B. 120
C. 150
D. 180
E. 240

[fskilnik@GMATH]

[Math Revolution GMAT math practice question]

How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime number?

A. 60
B. 120
C. 150
D. 180
E. 240

Very nice problem, Max. Congrats!
$$\left. \matrix{
\matrix{
{\underline {\,\,\,1\,\,\,} } \cr
{{\rm{yes/no}}} \cr

} \,\,\,\,\matrix{
{\underline {\,\,\,2\,\,\,} } \cr
{{\rm{yes/no}}} \cr

} \,\,\, \ldots \,\,\,\,\matrix{
{\underline {\,\,\,7\,\,\,} } \cr
{{\rm{yes/no}}} \cr

} \,\,\,\,\matrix{
{\underline {\,\,\,8\,\,\,} } \cr
{{\rm{yes/no}}} \cr

} \,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{2^8}\,\,{\rm{subsets}} \hfill \cr
\matrix{
{\underline {\,\,\,1\,\,\,} } \cr
{{\rm{yes/no}}} \cr

} \,\,\,\,\matrix{
{\underline {\,\,\,4\,\,\,} } \cr
{{\rm{yes/no}}} \cr

} \,\,\,\,\matrix{
{\underline {\,\,\,6\,\,\,} } \cr
{{\rm{yes/no}}} \cr

} \,\,\,\,\matrix{
{\underline {\,\,\,8\,\,\,} } \cr
{{\rm{yes/no}}} \cr

} \,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{2^4}\,\,{\rm{subsets}}\,\,{\rm{with}}\,\,{\rm{no}}\,{\rm{ - }}\,{\rm{primes}}\,\,\,\, \hfill \cr} \right\}\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {2^8} - {2^4} = {2^4}\left( {{2^4} - 1} \right) = 240$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

[Max@Math Revolution]

=>

It is easiest to use complementary counting. That is, count the number of subsets that contain no prime number and subtract it from the total number of subsets.

The number of subsets containing no prime number is the number of subsets of { 1, 4, 6, 8 }. Note that 1 is neither a prime number nor a composite number.

The number of subsets of {1,2,3,4,5,6,7,8} is 2^8 = 256.
The number of subsets of {1,4,6,8} is 2^4 = 16.
Thus, the number of subsets of {1,2,3,4,5,6,7,8} containing at least one prime number is 256 - 16 = 240.

Therefore, the answer is E.
Answer: E

[Scott@TargetTestPrep]

[Math Revolution GMAT math practice question]

How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime number?

A. 60
B. 120
C. 150
D. 180
E. 240

We can use the formula:

The number of subsets with at least one prime number = Total number of subsets - the number of subsets that have no prime numbers

The total number of subsets of a set with n elements is 2^n. Therefore, there are 2^8 subsets in the given set. Since the prime numbers are 2, 3, 5, and 7, the numbers in the set that are not primes are 1, 4, 6 and 8. The number of subsets these 4 numbers can create is 2^4.

Therefore, the number of subsets with at least one prime number is 2^8 - 2^4 = 256 - 16 = 240.

Answer: E.