M70-28

Official Solution:


Of the 27 participants in a talent show, 13 are singers, 14 are musicians, and 9 are dancers. If 4 people are both singers and dancers, 3 are both singers and musicians and 1 person is all three, how many people are both dancers and musicians?


A. 0
B. 1
C. 2
D. 3
E. 4


We’ll go for ALTERNATIVE because the numbers in the answers are easy to use.

Since we have three overlapping sets, we’ll draw a Venn diagram, and instead of going into redundant calculations, we’ll try the median answer (C) 2 and see whether it works out:

The third layer, consisting of all three groups, is made up of 1 person. This makes the number of people who are singers and musicians (but not dancers) 3 – 1 = 2, and the number of singer-dancers (but not musicians) 4 – 1 = 3. The number of people who only sing is 13 – 1 – 2 – 3 = 7. Now, if answer (C) is correct, the number of those who only dance is 9 – 3 – 1 – 1 = 4, and the number of those who are only musicians is 14 – 2 – 1 – 1 = 10. So the overall number of participants is: 7 + 3 + 1 + 2 + 10 + 1 + 4 = 28, instead of 27. (C) is eliminated.

We can just check a random answer, but note that the answer we need to check is actually a larger one in order to make both ‘just musicians’ and ‘just dancers’ groups smaller (and thus making the total number of participants smaller). Let’s see answer choice (D) 3:

As we’ve seen before, we have 1 person in all three groups; 2 who are singers and musicians (but not dancers); 3 singer-dancers (but not musicians); and 7 only sing. Now, if answer (D) is correct, the number of those who only dance is 9 – 3 – 1 – 2 = 3, and the number of those who are only musicians is 14 – 2 – 1 – 2 = 9. So the total number of participants is: 7 + 3 + 1 + 2 + 9 + 2 + 3 = 27. That’s our answer!


Answer: D