Question

In a class, 18 students opted Physics, 23 opted Chemistry and 24 students
opted Mathematics. Of these 13 opted both Chemistry and Mathematics,
12 opted both Physics and Chemistry and 11 opted Physics and
Mathematics. If 6 students opted all the three subjects, find how many
opted exactly one of the 3 subjects.​

Answer 1

Given:

Students that opted physics = 18

Students that opted chemistry = 23

Students that opted mathematics = 24

Students that opted both chemistry and mathematics = 13

Students that opted both physics and chemistry = 12

Students that opted both physics and mathematics = 11

Students that opted all the three subjects = 6

To find:

Number of students that opted exactly one subject.

Solution:

Let:

Mathematics = M

Physics = P

Chemistry = C

We know that n(M∩P∩C) = 6

n(M) = 24

n(P) = 18

n(C) = 23

n(M∩P) = 11

n(M∩C) = 13

n(C∩P) = 12

We know that:

n(M∪P∪C) = n(M) + n(P) +n(C) -n(M∩P)- n(M∩C) -n(C∩P)+ n(M∩P∩C)

n(M∪P∪C) = 24 + 18 + 23 - 11 - 13 - 12 + 6

n(M∪P∪C) = 35

Students that opted for only mathematics =n(M) -n(M∩P)- n(M∩C) + n(M∩P∩C)

24 - 11- 13 + 6 = 6

Students that opted for only physics = n(P) -n(M∩P) -n(C∩P)+ n(M∩P∩C)

= 18 - 11 - 12 + 6 = 1

Students that opted for only Chemistry =n(C) - n(M∩C) -n(C∩P)+ n(M∩P∩C)

=23- 13- 12 +6 = 4

Number of students that only opted exactly one subject = 6 + 1+ 4 = 11

The required answer is 11.