Question

A survey of 73 college students was taken to determine the musical styles they liked. Of​ those, 27 students listened to​ rock, 24 to​ classical, and 28 to jazz.​ Also, 16 students listened to rock and​ jazz, 13 to rock and​ classical, and 12 to classical and jazz.​ Finally, 10 students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region.

Answer 1

Attached is a Venn diagram of your problem.

Knowing how many likes all three will help. You know that 10 students like all three.

Rock and Jazz only:
16 like rock and jazz while 10 like all three. To get how many like jazz only, subtract 10 from 16.

16-10 = 6

Rock and Classical only:
13 like rock and classical while 10 like all three. To get how many like jazz only, subtract 10 from 13.

13-10 = 3

Jazz and classical only:
12 like jazz and classical while 10 like all three. To get how many like jazz only, subtract 10 from 12.

12-10 = 2

Now with that data you fill up the 4 intersecting areas. To get the outer, just remember that all areas within a circle should add up to the first assumption.
27 rock
24 classical
28 Jazz

All numbers in the rock circle should add up to 27.
All numbers in the classical circle should add up to 24.
All numbers in the Jazz circle should add up to 28.

Rock:
3+10+6+x = 27
19+x=27
x = 27-19
x= 8

Classical:
3+10+2+x = 24
15 + x = 24
x = 24-15
x = 9


Jazz:
10+6+2+x = 28
18 + x = 28
x = 28 - 18
x = 10

In summary: 8 liked only Rock, 9 liked only Classical, 10 liked only Jazz.