In a survey of the dining preferences of 54 dormitory students at the end of the spring semester, the following facts were discovered about Adam's Lunch AL, Pizza Tower PT, and the Dining Hall DH.
24 liked AL but not PT
20 liked AL only
37 liked AL
24 liked PT
22 liked DH
9 liked PT and AL but not DH
12 liked PT and DH

How many liked only DH?

A)10
B)6
C)74
D)18
E)22

Question

In a survey of the dining preferences of 54 dormitory students at the end of the spring semester, the following facts were discovered about Adam's Lunch AL, Pizza Tower PT, and the Dining Hall DH. ​
24 liked AL but not PT
20 liked AL only
37 liked AL
24 liked PT
22 liked DH
9 liked PT and AL but not DH
12 liked PT and DH
​
How many liked only DH?
​
A)10
B)6
C)74
D)18
E)22

Quizplus · Accepted Answer

To find the number of students who liked only DH, we can use a Venn diagram. Let A be the set of students who liked AL, B be the set of students who liked PT, and C be the set of students who liked DH.

From the given information, we can fill in the Venn diagram as follows:

- A: 37 total, 20 only liked AL, 9 liked PT and AL but not DH, so 37 - 20 - 9 = 8 liked AL and DH but not PT, and 37 - 20 - 8 = 9 liked AL and DH and PT. 
- B: 24 total, 24 only liked PT, 9 liked PT and AL but not DH, and 12 liked PT and DH, so 24 - 24 - 9 - 12 = -21, which doesn't make sense in the context of the problem. We made an error somewhere, so we can go back and check our work with respect to PT. We notice that 9 + 24 + 12 = 45, but there were only 54 total students in the survey. This means that there must be 9 students who didn't like any of the three options, so we can add a new set D to the Venn diagram to account for these students. 
- D: 9 total, representing the students who didn't like any of the three options.

With this correction, we can now fill in the Venn diagram as follows:

- A: 20 only liked AL, 8 liked AL and DH but not PT, 9 liked PT and AL but not DH, 9 liked AL and DH and PT, 37 total. 
- B: 24 only liked PT, 9 liked PT and AL but not DH, 12 liked PT and DH, 0 liked PT and DH but not AL, 45 total. 
- C: 22 liked DH, 8 liked AL and DH but not PT, 12 liked PT and DH, 9 liked AL and DH and PT, 51 total. 
- D: 9, representing the students who didn't like any of the three options.

To find the number of students who liked only DH, we look at the intersection of sets B and C, which is 12. Therefore, the answer is 12.

In a Survey of the Dining Preferences of 54 Dormitory