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# Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3

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Joined: 18 Oct 2009
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Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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30 Oct 2009, 21:32
5
51
00:00

Difficulty:

95% (hard)

Question Stats:

28% (02:15) correct 72% (02:16) wrong based on 774 sessions

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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

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Posts: 58434
Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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05 Feb 2012, 03:47
2
2
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:

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.

Detailed analysis of this concept is here: http://gmatclub.com/forum/formulae-for- ... ml#p729340

subhajeet wrote:
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.

Attachment:

Union_3sets.gif [ 11.63 KiB | Viewed 8318 times ]

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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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11 Jul 2010, 12:02
15
6
This can be explained easily with a Venn Diagram.
Attachment:

Venn Diagram.jpg [ 23.96 KiB | Viewed 31074 times ]

Piano: x
Guitar: y
Violin: z
Piano and Guitar: a
Piano and Violin: b
Guitar and Violin: c

To find: a+b+c + 3

Given Conditions:

x+a+b+3 = 10
a+b = 7-x (1)

a+y+3+c = 11
a+c = 8-y (2)

b+c+z+3 = 14
b+c = 11-z (3)

x+y+z = 20

Adding 1, 2 and 3, we get :

a+b+b+c+a+c = 7-x+8-y+11-z = 26-(x+y+z) = 26-20 = 6

2(a+b+c) = 6
(a+b+c) = 3

Final answer: Number of people who play 2 instruments = Number of people who play ONLY two + # who play 3 = 3+3 = 6

##### General Discussion
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Joined: 18 Aug 2009
Posts: 251
Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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30 Oct 2009, 23:24
5
3
Here is one approach to get B... shoot!!
3 people play all three instruments. So they got counted in 10 of Piano, 11 of Guitar and 14 of Violin.
So people who play 1 or 2 instruments is 35-3x3=26
Given 20 people play 1 instrument only... people who play 2 instruments= 26-20=6

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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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31 Oct 2009, 11:31
gmattokyo wrote:

Here is one approach to get B... shoot!!
3 people play all three instruments. So they got counted in 10 of Piano, 11 of Guitar and 14 of Violin.
So people who play 1 or 2 instruments is 35-3x3=26
Given 20 people play 1 instrument only... people who play 2 instruments= 26-20=6

That bold st. is the key. The 3 people who play all three instruments is included in each of these, "10 play piano, 11 play guitar, 14 play violin" so it has to be subtracted from each.

It should have been:

Let single piano players = p
Let single guitar players = g
Let single violin players = v

Given, p + 3 + g + 3 + v + 3 + 20 = 10 + 11 + 14.
Hence, p + g + v = 35 - 29 = 6
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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16 Mar 2010, 00:48
4
Jivana wrote:
gmattokyo wrote:
Here is one approach to get B... shoot!!
3 people play all three instruments. So they got counted in 10 of Piano, 11 of Guitar and 14 of Violin.
So people who play 1 or 2 instruments is 35-3x3=26
Given 20 people play 1 instrument only... people who play 2 instruments= 26-20=6

That bold st. is the key. The 3 people who play all three instruments is included in each of these, "10 play piano, 11 play guitar, 14 play violin" so it has to be subtracted from each.

It should have been:

Let single piano players = p
Let single guitar players = g
Let single violin players = v

Given, p + 3 + g + 3 + v + 3 + 20 = 10 + 11 + 14.
Hence, p + g + v = 35 - 29 = 6

3 people play all three instruments. So they got counted in 10 of Piano, 11 of Guitar and 14 of Violin.
ppl playing Piano & Guitar but not Violin counted twice; in 10 and 11
ppl playing Guitar & Violin but not Piano counted twice; in 11 and 14
ppl playing Violin & Piano but not Guitar counted twice; in 14 and 10

ie ppl playing only two instruments are counted twice if we add 10 , 11 , 14

ie 10 + 11 + 14 = 20 + 2B + 3 x 3
2B = 6
B = 3 ,

The Qn is how many ppl play two instruments

= ppl playing only two instruments + ppl playing 3 instruments
3 + 3 = 6

-V
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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16 Mar 2010, 08:19
1
slingfox wrote:
Can someone tell explain how they go about solving this one?

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

Lets' say PG+GV+VP = S where PG is intersection of piano and guitar, GV is intersection of guitar and violin and VP is intersection of violin and piano

No. of players who play at least one instrument = 10+11+14 - S +3 or (P+G+V-(PG+GV+VP)+PGV)
=38 - S

No. of players who play at least one instrument also = 20+s-6 or (No.of people who play only one instrument + PG+GV+VP - 2 * PGV)
14+s = 38 - s
=>2S = 24 => s = 12

No. of people who play only 2 instruments = PG+GV+VP - 2 * PGV = 12 - 2*3 = 6

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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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25 Jul 2010, 05:25
2
1
One instruments = A+B+C - 2(AB+BC+AC) +3ABC

20 = 11+14+10 -2X +9

=> X= 17

Two instruments = AB+BC+AC - 3ABC = 17 -9 =6

Hence B
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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27 Aug 2011, 12:22
1
1
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)
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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06 Sep 2011, 21:57
MBAhereIcome wrote:
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 7931 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 violin, 3  [#permalink]

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05 Feb 2012, 03:19
VeritasPrepKarishma wrote:
MBAhereIcome wrote:
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.
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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22 Aug 2012, 09:49
1
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 violin, 3  [#permalink]

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15 Aug 2013, 15:18
1
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
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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15 Aug 2013, 20:05
VeritasPrepKarishma wrote:
MBAhereIcome wrote:
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???
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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15 Aug 2013, 22:26
jaituteja wrote:
VeritasPrepKarishma wrote:
MBAhereIcome wrote:
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.
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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01 Aug 2015, 21:17
10 piano
11 guitar
14 violin
Total is 35
3 play all 3 instruments
20 play just one

That means that of the 35 counted, 20 of them are not duplicates.
15 are duplicates. Of the 15, 3 play all three instruments, so each person is counted 3 times so 9 total counts when they should be counted only 3 times. 35-20-9 = remaining number = 6, each of these include people who play two instruments, so they're repeated twice each. So, there are three people who play all two instruments. 3 people play all three + 3 people play two = 6 people that play at least 2 instruments (at least as opposed to only 2 instruments).
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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19 Mar 2016, 04:27
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

A: No of people only playing piano
B: No of people only playing Guitar
C: No of people only playing Violin
D: No of people playing only Piano and Guitar
E: No of people playing only piano and Violin
F: No of people laying only Violin and Guitar
G: No of people playing all three

35 = A+B+C+D+E+F+G
35=20+D+E+F+3
D+E+F=12
So No. of people playing ONLY 2 instruments is 12.
and No. of people playing 2 instrument = No of people playing ONLY 2 instrument + No. of people playing ONLY 3 instrument, which i equal to 12+3=15.
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Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3  [#permalink]

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04 Jun 2019, 13:46
Had hard times with it - not anymore - take a look at explanation by Bunuel here https://gmatclub.com/forum/formulae-for ... 69014.html

Now any overlapping set question is easy - no need to remember the formulas - just remember the picture.
Re: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3   [#permalink] 04 Jun 2019, 13:46
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