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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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13 Dec 2018, 06:12
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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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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Manager Joined: 14 Jun 2018 Posts: 217 Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe [#permalink] Show Tags 13 Dec 2018, 07:29 1 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 GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 935 Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe [#permalink] Show Tags 13 Dec 2018, 08:55 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. _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our high-level "quant" preparation starts here: https://gmath.net Senior Manager Joined: 27 Dec 2016 Posts: 309 Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe [#permalink] Show Tags 13 Dec 2018, 19:23 Could someone please explain how we are calculating 2^8 as the total subsets? Intern Joined: 02 Dec 2018 Posts: 3 How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe [#permalink] Show Tags 14 Dec 2018, 01:57 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) Posted from my mobile device Intern Joined: 02 Dec 2018 Posts: 3 Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe [#permalink] Show Tags 14 Dec 2018, 02:00 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) Posted from my mobile device GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 935 Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe [#permalink] Show Tags 14 Dec 2018, 12:08 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. _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our high-level "quant" preparation starts here: https://gmath.net Intern Joined: 31 Mar 2017 Posts: 8 Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe [#permalink] Show Tags 16 Dec 2018, 16:35 2^8-2^4= 256 - 16 = 240 Posted from my mobile device Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8017 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe [#permalink] Show Tags 16 Dec 2018, 18:39 => 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 _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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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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14 Mar 2019, 07:18
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?

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

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.

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Re: How many subsets of {1,2,3,4,5,6,7,8} contain at least one prime numbe   [#permalink] 14 Mar 2019, 07:18
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