In high-school 135 freshmen were interviewed. Thirty five of them took PE, 42 took BIO, 60 took ENG, 10 took PE and BIO, 15 took PE and ENG, 7 took BIO and ENG, 5 took all three subjects. a How many students took none of the three subjects

25

Step-by-step explanation:

According to set theory, the number of element in a set is its cardinality. Let n (P), n (B) and n (E) represent those that took physics, Bio and Eng respectively.

Given n (U) = 135 where U is the universal set.

n (P) = 35

n (B) = 42

n (E) = 60

n (P∩B) = 10

n (P∩E) = 15

n (B∩E) = 7

n (P∩B∩E) = 5

According to set theory, n (U) = n (PUBUE) + n (PUBUE) ' where n (PUBUE) ' are students took none of the three subjects.

Before we can get n (PUBUE) ' we need to get n (PUBUE). Using the relationship below;

n (PUBUE) = n (P) + n (B) + n (E) - n (P∩B) - n (P∩E) - n (B∩E) + n (P∩B∩E)

n (PUBUE) = 35+42+60-10-15-7+5

n (PUBUE) = 110

since n (U) = n (PUBUE) + n (PUBUE) '

135 = 110+n (PUBUE) '

n (PUBUE) ' = 135-110

n (PUBUE) ' = 25

This shows that 25 students took none of the three subjects