In a class of 32 students, 17 study music and 20 study art. What is the least number of students who study both subjects?

• A. 5
• B. 7
• C. 10
• D. 12

Answer: (A)

To determine the least number of students who study both music and art, we can use the principle of inclusion-exclusion.

First, we need to know the total number of students in the class, which is 32. Next, we add the number of students who study music (17) to the number of students who study art (20):

[

17 + 20 = 37.

]

Since there are only 32 students in total, this sum includes some students counted twice—those who study both subjects. To find this overlap, we subtract the total number of students from the combined total of music and art students:

[

37 - 32 = 5.

]

This means that at least 5 students must be studying both music and art. Thus, the least number of students who study both subjects is 5, which corresponds to the correct response.

This analysis clarifies why the correct answer is 5, as it is derived from the constraints of the total number of students and the sums of those studying each subject, ensuring no student is unaccounted for in the total.