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I worked on this using Venn Diagram. Got the answer 6 in one shot. The answer is how many play 2 instruments so we need to count the people who also play 3 instruments ( given already 3).

those who play single instrument = 20 those who don't play a single instrument (play 2 or 3) = 35-20 = 15

those who play 2 instruments = 15-3 = 12 (3 play all three instruments)

The 35 that you counted: 10 + 11 + 14 double counts the number of people who play only 2 instruments and triple counts the number of people who play all 3 instruments. The circle of 10 of piano includes the red and the green region. The circle of 11 of Guitar includes the red and green region too. Similarly, the circle of 14 of violin includes the red and the green region too. So each red region is counted twice and the green region is counted 3 times.

Attachment:

Ques2.jpg [ 14.89 KiB | Viewed 4741 times ]

So 35 - 20 = 15 15 - 3 = 12 (Subtract once the triple counted green region. Now everything is double counted) 12/2 = 6 = Total number of people who play 2 or 3 instruments.
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink]

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05 Feb 2012, 01:18

slingfox wrote:

Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3 B. 6 C. 9 D. 12 E. 15

Total folks = 10 + 11 + 14 = 35 20 Play only one instrument. So no of people playing two or more instruments = 35 - 20 = 15. From the venn diagrams we know that 15 represents the intersection part between Piano, Guitar and Violin. In the venn diagram, people having all the three capabilities are counted thrice. Hence people playing exactly two instruments = 15 - 3 (3) = 6

Hence B. Direct formulas may not be applicable all the time. Focus on which area is being counted and how many times.

Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink]

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05 Feb 2012, 01:47

The way I did it was add up all the instruments together, so 10 + 11 +14 = 35. I knew 3 people played all 3, so I subtract 3 from each of the instruments, so 7+8+11 = 26. I know this is the amount of instruments played by a person who plays 1 or 2 instruments. Since the amount of people that play 1 instrument is 20, you subtract all the single instrument from the total. 26-20=6, so 6 instrument. We know that people in this group play 2 instruments, so there is 3 people playing 6 instruments. 3 people who plays exactly 2 instruments + 3 people who plays exactly 3 = 6 people who play 2 or more instruments.

those who play single instrument = 20 those who don't play a single instrument (play 2 or 3) = 35-20 = 15

those who play 2 instruments = 15-3 = 12 (3 play all three instruments)

The 35 that you counted: 10 + 11 + 14 double counts the number of people who play only 2 instruments and triple counts the number of people who play all 3 instruments. The circle of 10 of piano includes the red and the green region. The circle of 11 of Guitar includes the red and green region too. Similarly, the circle of 14 of violin includes the red and the green region too. So each red region is counted twice and the green region is counted 3 times.

Attachment:

Ques2.jpg

So 35 - 20 = 15 15 - 3 = 12 (Subtract once the triple counted green region. Now everything is double counted) 12/2 = 6 = Total number of people who play 2 or 3 instruments.

Hi, Dont you think the answer should be A i.e. 3 since the question asks for how many people who play 2 instruments and not atleast 2 instruments.

Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3 B. 6 C. 9 D. 12 E. 15

To elaborate more.

Look at the diagram below:

Attachment:

Union_3sets.gif [ 11.63 KiB | Viewed 4625 times ]

To solve this question one should fundamentally understand two things: 1. What does the question ask: "How many play 2 instruments?" So, we should find the sum of the sectors 1, 2, 3, and 4. Notice that those who play two instruments include also those who play all three instruments, (sector 4);

2. What happens when we sum all three groups, 10 piano players, 11 guitar players and 14 violin players? When we add these three groups, we'll get 10+11+14=35 but some sections are counting more than once in this number: sections 1, 2, and 3 are counted twice and section 4 thrice. Now, if we subtract those who play only one instrument (inner white sections on the diagram), we'll get 35-20=15, so twice sections 1, 2, and 3 plus thrice section 4 equals to 15.

Since, 15 counts section 4, those who play all the instruments, thrice then of 15-3=12 counts these section twice. So, now 12 counts all sections 1, 2, 3 and 4 twice. We need to count them once thus divide this number by 2 --> 12/2=6 play 2 instruments.

Hi, Dont you think the answer should be A i.e. 3 since the question asks for how many people who play 2 instruments and not atleast 2 instruments.

No, if it were the case question would ask: "how many play EXACTLY 2 instruments?" How many play 2 instruments, means how many play at least 2 instruments, hence this group includes also those who play all 3 instruments.

Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink]

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06 Feb 2012, 01:20

Bunuel wrote:

slingfox wrote:

Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3 B. 6 C. 9 D. 12 E. 15

To elaborate more.

Look at the diagram below:

Attachment:

Union_3sets.gif

To solve this question one should fundamentally understand two things: 1. What does the question ask: "How many play 2 instruments?" So, we should find the sum of the sectors 1, 2, 3, and 4. Notice that those who play two instruments include also those who play all three instruments, (sector 4);

2. What happens when we sum all three groups, 10 piano players, 11 guitar players and 14 violin players? When we add these three groups, we'll get 10+11+14=35 but some sections are counting more than once in this number: sections 1, 2, and 3 are counted twice and section 4 thrice. Now, if we subtract those who play only one instrument (inner white sections on the diagram), we'll get 35-20=15, so twice sections 1, 2, and 3 plus thrice section 4 equals to 15.

Since, 15 counts section 4, those who play all the instruments, thrice then of 15-3=12 counts these section twice. So, now 12 counts all sections 1, 2, 3 and 4 twice. We need to count them once thus divide this number by 2 --> 12/2=6 play 2 instruments.

Hi, Dont you think the answer should be A i.e. 3 since the question asks for how many people who play 2 instruments and not atleast 2 instruments.

No, if it were the case question would ask: "how many play EXACTLY 2 instruments?" How many play 2 instruments, means how many play at least 2 instruments, hence this group includes also those who play all 3 instruments.

Refer to the link above for more on this issue.

Hope it helps.

Bunnel: Thanks for the explanation. But it becomes a bit confusing here. Looks like i have to extra vigilent for these type of statements.

Of a group of people, 10 play piano, 11 play guitar [#permalink]

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22 Aug 2012, 08:49

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Question wordings are very poor or at least very ambiguous. "How many play 2 instruments?" can be inferred in 2 ways (both of which are right) which creates unnecessary confusion 1) Play Exactly 2 instruments 2) Play at least 2 instruments
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink]

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15 Aug 2013, 13:58

Yes its language is not correct someone should edit it "how many play at least two instrument" or Change the OA to - A.

I also got 3

when i saw 26% correct answers i thought its a tough problem but later i discovered this issue, which everyone else has also reported.
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink]

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15 Aug 2013, 14:18

1

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slingfox wrote:

Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3 B. 6 C. 9 D. 12 E. 15

guitar = 11 piano = 10 violin = 14

1 or 2 instruments player = 10-3 +11-3 +14-3 = 35-9 = 26 1 instruments player = 20 so 2 instruments player = 26-20 = 6 =B
_________________

those who play single instrument = 20 those who don't play a single instrument (play 2 or 3) = 35-20 = 15

those who play 2 instruments = 15-3 = 12 (3 play all three instruments)

The 35 that you counted: 10 + 11 + 14 double counts the number of people who play only 2 instruments and triple counts the number of people who play all 3 instruments. The circle of 10 of piano includes the red and the green region. The circle of 11 of Guitar includes the red and green region too. Similarly, the circle of 14 of violin includes the red and the green region too. So each red region is counted twice and the green region is counted 3 times.

Attachment:

Ques2.jpg

So 35 - 20 = 15 15 - 3 = 12 (Subtract once the triple counted green region. Now everything is double counted) 12/2 = 6 = Total number of people who play 2 or 3 instruments.

The question only states that how many play 2 instruments. Nowhere it is mentioned that it asks for only 2 instruments or atleast 2 instruments???
_________________

MODULUS Concept ---> http://gmatclub.com/forum/inequalities-158054.html#p1257636 HEXAGON Theory ---> http://gmatclub.com/forum/hexagon-theory-tips-to-solve-any-heaxgon-question-158189.html#p1258308

those who play single instrument = 20 those who don't play a single instrument (play 2 or 3) = 35-20 = 15

those who play 2 instruments = 15-3 = 12 (3 play all three instruments)

The 35 that you counted: 10 + 11 + 14 double counts the number of people who play only 2 instruments and triple counts the number of people who play all 3 instruments. The circle of 10 of piano includes the red and the green region. The circle of 11 of Guitar includes the red and green region too. Similarly, the circle of 14 of violin includes the red and the green region too. So each red region is counted twice and the green region is counted 3 times.

Attachment:

Ques2.jpg

So 35 - 20 = 15 15 - 3 = 12 (Subtract once the triple counted green region. Now everything is double counted) 12/2 = 6 = Total number of people who play 2 or 3 instruments.

The question only states that how many play 2 instruments. Nowhere it is mentioned that it asks for only 2 instruments or atleast 2 instruments???

I understand the confusion. But the language of sets is very mathematical and literal. Say, if I ask you, whether you play an instrument and you have to answer in yes or no, you will say yes even if you play 2/3/4 instruments. Similarly, if I ask you whether you play two instruments, you will answer yes even if you play 3/4 instruments.

No of people who play two instruments includes number of people who play more than 2 too since these people certainly do play 2 instruments (and they play some more).

If 'at least' or 'only' is mentioned, we will consider it accordingly.
_________________

Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink]

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18 Aug 2013, 05:09

PiyushK wrote:

Yes its language is not correct someone should edit it "how many play at least two instrument" or Change the OA to - A.

I also got 3

when i saw 26% correct answers i thought its a tough problem but later i discovered this issue, which everyone else has also reported.

The question never said exactly 2 instrument; had it done so your answer would have been correct, but the question said people who play two instrument which might include 3 people. hence the solution is exactly 2 once and all three once; 3+3=6
_________________

--It's one thing to get defeated, but another to accept it.

Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink]

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18 Aug 2013, 10:36

slingfox wrote:

Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3 B. 6 C. 9 D. 12 E. 15

play only piano 10-3=7 Play only guitar 11-3=8 Play only violin 14-3=11 so, 20=7+8+11-2overlapping set 2overlapping set=26-20=6 so, the best answer is (B) Posted by Abdul Mannan Mian

Re: Of a group of people, 10 play piano, 11 play guitar, 14 play [#permalink]

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14 Sep 2013, 02:09

saxenaashi wrote:

slingfox wrote:

Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument. How many play 2 instruments?

A. 3 B. 6 C. 9 D. 12 E. 15

Total folks = 10 + 11 + 14 = 35 20 Play only one instrument. So no of people playing two or more instruments = 35 - 20 = 15. From the venn diagrams we know that 15 represents the intersection part between Piano, Guitar and Violin. In the venn diagram, people having all the three capabilities are counted thrice. Hence people playing exactly two instruments = 15 - 3 (3) = 6

Hence B.

Direct formulas may not be applicable all the time. Focus on which area is being counted and how many times.

Here "6" that you have got is counted twice i.e. 2(Exactly two) = 6 Therefore 3 will be exactly two..

Hence if question asks 2 instruments then it will be Exactly 2 + All = 3 + 3 = 6 people..

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