- Of 60 students in a class studying maths, physics, and chemistry, anyone studying maths also studied physics. No one studied both maths and chemistry. 16 students studied both physics and chemistry. - The number studying exactly one subject must be greater than the number studying more than one subject. - Using this information in diagrams and equations, the minimum number of students who could have studied chemistry only is 0, and the maximum is 44.

1. Set Theory

2. Set Theory
Of 60 students in a class, anyone who has chosen to study maths elects to do physics
as well. But no one does maths and chemistry, 16 do physics and chemistry. All the
students do at least one of the three subjects and the number of people who do
exactly one of the three is more than the number who do more than one of the three.
What are the maximum and minimum number of people who could have done
Chemistry only?
(a) 40, 0 (b) 28, 0
(c) 38, 2 (b) 44, 0

3. Set Theory
Let us outline the diagram. Anyone who does Maths does Physics
also.
Maths is a subset of Physics. Now, let us build on this.
Physics
Maths
60

4. Set Theory
No one does maths and chemistry, 16 do physics and chemistry.
Number outside is 0 as all the students do at least one of the three
subjects
Physics
Maths
60
0
16
a
b
c

5. Set Theory
a + b + c +16 = 60, or a + b + c = 44
The number of people who do exactly one of the three is more than
the number who do more than one of the three. => a + b > c + 16
Physics
Maths
60
0
16
a
b
c

6. Set Theory
So, we have a + b + c = 44 and a + b > c + 16
We need to find the maximum and minimum possible values of b.
Let us start with the minimum. Let b = 0, a + c = 44. a > c + 16. We
could have
a = 40, c = 4. So, b can be 0.
Now, thinking about the maximum value. b = 44, a = c = 0 also works.
So, minimum value = 0, maximum value = 44.
Answer choice (d)

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