In a class of 175 students the following data shows the number of students opting one or more subjects.
Question:

In a class of 175 students the following data shows the number of students opting one or more subjects. Mathematics 100; Physics 70; Chemistry 40; Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and Chemistry 23; Mathematics, Physics and Chemistry 18. How many students have offered Mathematics alone?

(a) 35

(b) 48

(c) 60

(d) 22

(e) 30

Solution:

(c) 60

Let M, P and C denote the sets of students who have opted for mathematics, physics, and chemistry, respectively.

Here,

$n(M)=100, n(P)=70$ and $n(C)=40$

Now,

$n(M \cap P)=30, n(M \cap C)=28, n(P \cap C)=23$ and $n(M \cap P \cap C)=18$

Number of students who opted for only mathematics:

$n\left(M \cap P^{\prime} \cap C^{\prime}\right)=n\left\{M \cap(P \cup C)^{\prime}\right\}$

$=n(M)-n\{M \cap(P \cup C)\}$

$=n(M)-n\{(M \cap P) \cup(M \cap C)\}$

$=n(M)-\{n(M \cap P)+n(M \cap C)-n(M \cap P \cap C)\}$

$=100-(30+28-18)$

$=60$

Therefore, the number of students who opted for mathematics alone is 60