Max@Math Revolution wrote:[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.

=>

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