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

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

step-by-step explanation: where's the phrase?

to have more of what you want like money

step-by-step explanation:

save it by not spending on unnecessary stuff and have more