Question:
In a survey of 25 students it was found that 15 had taken Maths, 12
had taken Physics and 11 had taken Chemistry, 5 had taken Maths
and chemistry, 9 had taken Maths and Physics, 4 had taken Physics
and Chemistry and 3 had taken all the three subjects. Find the number
of students that had taken:
i) Only Chemistry
ii) Only Maths
iii) Only Physics
iv) Physics and Chemistry but not Maths
v) Maths and Physics but not Chemistry
vi) Only one of the subject
vii) At least one of the subjects
viii) None of the subjects
Answer:
Let M: Set of students who have taken Maths
P: Set of students who have taken Physics
C: Set of students who have taken Chemistry
Given,
Total students n(U) = 25
n(M) = 15, n(P) = 12, n(C) = 11
n(M ∩ C) = 5, n(P ∩ C) = 4, n(M ∩ P) = 9,
n(M ∩ P ∩ C) = 3
1. Number of students taking only Chemistry = n(C - (M ∪ P))
= n(C) - n(C ∩ (M ∪ P))
= n(C) - [n(C ∩ M) + n(C ∩ P) - n((C ∩ M) ∩ (C ∩ P)) ]
= n(C) - n(C ∩ M) - n(C ∩ P) + n(C ∩ M ∩ P)
= 11 - 5 - 4 + 3
= 14 - 9
= 5
2. Number of students taking only Maths = n(M - (P ∪ C))
= n(M) - n(M ∩ (P ∪ C))
= n(M) - [n(M ∩ P) + n(M ∩ C) - n((M ∩ P) ∩ (M ∩ C)) ]
= n(M) - n(M ∩ P) - n(M ∩ C) + n(M ∩ P ∩ C)
= 15 - 9 - 5 + 3
= 18 - 14
= 4
3. Number of students taking only Physics = n(P - (M ∪ C))
= n(P) - n(P ∩ (M ∪ C))
= n(P) - [n(P ∩ M) + n(P ∩ C) - n((P ∩ M) ∩ (P ∩ C)) ]
= n(P) - n(P ∩ M) - n(P ∩ C) + n(P ∩ M ∩ C)
= 12 - 9 - 4 + 3
= 15 - 13
= 2
4. Number of students taking Physics and Chemistry but not Maths = n((P ∩ C) - M)
= n(P ∩ C) - n(P ∩ M ∩ C)
= 4 - 3
= 1
5. Number of students taking Maths and Physics but not Chemistry = n((M ∩ P) - C)
= n(M ∩ P) - n(P ∩ M ∩ C)
= 9 - 3
= 6
6. Number of students taking only one subject = n((only M) + (only P) + (only C))
= n(only M) + n(only P) + n(only C)
= 4 + 2 + 5
= 11
7. Number of students taking at least one subject = n(M ∪ P ∪ C)
= n M) + n(P) + n(C) - n(M ∩ P) - n(P ∩ C) - n(M ∩ C) + n(M ∩ P ∩ C)
= 15 + 12 + 11 - 9 - 4 - 5 + 3
= 41 - 18
= 23
8. Number of students taking none of three subject = 25 - n(M ∪ P ∪ C)
= 25 - 23
= 2
Not what you are looking for? Go ahead and submit the question, we will get back to you.
