Target Test Prep GMAT Quant Challenge Problem 2

Challenge Question

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In a certain 120-person orchestra, each musician plays one or more of the following musical instruments: the piano, the violin, or the tuba. A total of 50 musicians play the violin, 70 musicians play the piano, and 60 musicians play the tuba. If 30 musicians play exactly two of the instruments, how many musicians play exactly all three of the instruments?

10

13

14

15

17

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Solution:

We're looking for the number of musicians who play exactly three instruments.

⇒ Total # of Unique Elements = # in (Group A) + # in (Group B) + # in (Group C) – # in (Groups of Exactly Two) – 2[#in (Group of Exactly Three)] + # in (Neither)

⇒ Let T = # in (Group of Exactly Three)

⇒ 120 = 50 + 70 + 60 – 30 – 2(T) + 0

⇒ 120 = 150 – 2T

⇒ 2T = 30

⇒ T= 15

Thus, 15 people play exactly all three instruments. Notice that since each musician must play one or more of the three instruments, the number of people in the Neither region is zero.

Correct answer:D