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

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.

Answer: B.

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



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.

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

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.



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.

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.

Given: Of a group of people, 10 play piano, 11 play guitar, 14 play violin, 3 play all the instruments, 20 play only one instrument.

Asked: How many play 2 instruments?



Total = Piano + Guitar + Violin - Two - 2*All
Total = 10 + 11 + 14 - Two - 2*3 = One only + Two + All = 20 + Two + 3
29 - Two = 23 + Two
Two = (29-23)/2 = 3

Number of people playing only 2 instruments = Two = 3
Number of people playing at least 2 instruments = Two + All = 3 + 3 = 6

IMO B

Shouldn't the question say that how many play 2 or more instruments?

Create a formular for those who play only one instrument



P + G + V -2PnG-2PnV-2GnV + 3PnGnV= 20

10+11+14-2(a+b+c)+9=20

-2(a+b+c) = -24

a+b+c = 12


But in counting those who play 2 instruments those who play 3 have been counted thrice so remove 2 which is 6

12-6 = 6


Answer B