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

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How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe  [#permalink]

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New post 13 Dec 2018, 05:12
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[Math Revolution GMAT math practice question]

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

\(A. 60\)
\(B. 120\)
\(C. 150\)
\(D. 180\)
\(E. 240\)

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Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe  [#permalink]

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New post 13 Dec 2018, 06:29
1
Total subsets = 2^8
Out of all the nos , 4 are prime and 4 are non prime.
Subsets without any prime = 2^4
Subsets with at least one prime = 256-16 = 240
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Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe  [#permalink]

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New post 13 Dec 2018, 07:55
MathRevolution 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?

\(A. 60\)
\(B. 120\)
\(C. 150\)
\(D. 180\)
\(E. 240\)

(This solution is similar to pandeyashwin´s above, but I guess it makes the reasoning a bit more explicit!)

\(\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\)


Regards,
Fabio.
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Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe  [#permalink]

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New post 13 Dec 2018, 18:23
Could someone please explain how we are calculating 2^8 as the total subsets?
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How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe  [#permalink]

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New post 14 Dec 2018, 00:57
1
csaluja wrote:
Could someone please explain how we are calculating 2^8 as the total subsets?


Take a set A containing only 1 element in it; A={a}
Total subsets possible=2; the set itself and the null set

Take another set B with two elements in it
B={a, s}
Total subsets={a},{s},{a,s} and {}
Therefore, 4=(2^2) subsets possible

You can check it for any other small no. So, we can say that total subsets for a set containing n elements is (2^n)

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Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe  [#permalink]

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New post 14 Dec 2018, 01:00
csaluja wrote:
Could someone please explain how we are calculating 2^8 as the total subsets?


Take a set A containing only 1 element in it; A={a}
Total subsets possible=2; the set itself and the null set

Take another set B with two elements in it
B={a, s}
Total subsets={a},{s},{a,s} and {}
Therefore, 4=(2^2) subsets possible

You can check it for any other small no. So, we can say that total subsets for a set containing n elements is (2^n)

Posted from my mobile device
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Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe  [#permalink]

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New post 14 Dec 2018, 11:08
csaluja wrote:
Could someone please explain how we are calculating 2^8 as the total subsets?

Hi csaluja !

Let me add to ShubhamAjmera95 ´s correct explanation:

Each element that belongs to the set X={1,2,..,7,8} wil be present ("yes") or will not be present ("no") in any given subset of the set X.

Therefore there are 2 possibilities for 1 ("yes","no") and for each of them, there are two possibilities for 2 ("yes", "no"), and so on.

The sequential application of the Multiplicative Principle goes:

2 (possibilities for 1, "yes" or "no") x 2 (idem for 2) x ... x 2 (idem for 7) x 2 (idem for 8) gives 2^8, the number of possible subsets of X.

Some explicit examples of the reasoning explained above:

"no" , "no" , "no" , ... , "no" ::: gives the null (void) set, that is, the set with no elements.
"yes" , "no" , "no" , ... , "no" ::: gives {1}
"yes" , "no" , "yes" , "no" , ... , "yes", "no" ::: gives {1, 3, 5, 7}
"yes" , "yes" , "yes" , ... , "yes", "yes" ::: gives X itself

I hope things got clearer.

Regards,
Fabio.
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Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe  [#permalink]

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New post 16 Dec 2018, 15:35
2^8-2^4= 256 - 16 = 240

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Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe  [#permalink]

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New post 16 Dec 2018, 17:39
=>

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
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Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe &nbs [#permalink] 16 Dec 2018, 17:39
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