Hi~ ^-^ please help me with this problem

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?
b) How many students took PE but not BIO?

a) 25 students took none of the three

b) 25 students took PE but not BIO

Step-by-step explanation:

a)

To find how many students took none, we can use the following formula (union of three events) to find the number of students that took PE or BIO or ENG:

P(PE or BIO or ENG) = P(PE) + P(BIO) + P(ENG) - P(PE and BIO) - P(PE and ENG) - P(BIO and ENG) + P(PE and BIO and ENG)

P(PE or BIO or ENG) = 35 + 42 + 60 - 10 - 15 - 7 + 5 = 110

110 students took any of the three, and the total students is 135, so we have 135 - 110 = 25 students that took none of the three.

b)

To find the students that took PE but not BIO, we can just subtract the students that took PE and BIO from the students that took PE:

P(PE and not BIO) = P(PE) - P(PE and BIO) = 35 - 10 = 25 students

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step-by-step explanatio89