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# Of a group of people, 10 play piano, 11 play guitar, 14 play

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Manager
Joined: 01 Mar 2014
Posts: 128
Schools: Tepper '18
Re: Of a group of people, 10 play piano, 11 play guitar, 14 play  [#permalink]

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19 Mar 2016, 09:51
whiplash2411 wrote:
fatihaysu wrote:
I disagree about that. We cant add 3 students because these 3 students play all of enstrument. We need the students who only play 2 enstruments.

The right answer should be 3 not 6. Any opinion why are we adding 3 to a+b+C

Please read the question. It asks for the number of people who play 2 instruments, not the number of people who play ONLY two instruments. The people who play 3 instruments obviously play two instruments as well, and hence can be counted in the group that plays two instruments. Had the question asked for people who play only two instruments, then, yes, the answer would be 3.

This explains it...!! careless mistake..
Retired Moderator
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play  [#permalink]

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05 Nov 2016, 08:23
Method 1:
Using formula: Only one (if exactly two is given) = All Single – 2(Exactly two) – 3(All Three)
$$20=10+11+14-2x-3(3) =>20=35-2x-9 =>2x=6=>x=3$$
∴ At least two = Exactly 2 + all three = 3+3 = 6

Method 2:
Using formula: Only one (if doubles is given) = All Single – 2(All Doubles) + 3(All Three)

$$20=10+11+14-2x+3(3) =>20=35-2x+9 =>2x=24=>x=12$$
∴ At least two = All doubles – 2 all three = 12-2(3) = 6

NB: The ‘All doubles’ already included ‘all 3’ in three times. So need to deduct ‘all 3’ two times to get the ‘at least two’ value, i.e., exactly two and exactly three.
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Hasan Mahmud

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Joined: 28 Jan 2018
Posts: 54
Location: Netherlands
Concentration: Finance
GMAT 1: 710 Q50 V36
GPA: 3
Re: Of a group of people, 10 play piano, 11 play guitar, 14 play  [#permalink]

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07 Feb 2018, 08:54
Here is my solution, it doesn't involve venn diagram:

lets call number of people play EXACTLY 2 instruments is x
People who play Piano is A
People who play Guitar is B
People who play Violin is C
then we got:
Number of people = A + B + C - x - 2*3 (1)
Also, we know that
Number of people = (People play exactly 1 instrument) + (People play exactly 2 instrument) + (People play exactly 3 instrument)
Number of people = 20 + x + 3 (2)

From both (1) and (2) we got the following equation: 29 - x = 23 + x, therefore x = 3

The question asked number of people play 2 instruments, not EXACTLY 2 instruments, so we need to add the number of people play 3 instrument

Final answer = 3+3 = 6 => B
Re: Of a group of people, 10 play piano, 11 play guitar, 14 play &nbs [#permalink] 07 Feb 2018, 08:54

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