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

Answer 1

(c) 60
Let M, P and C denote the sets of students who have opted for mathematics, physics, and chemistry, respectively.
Here,
$n\left(M\right)$ = 100, $n\left(P\right)$ = 70 and $n\left(C\right)$ = 40
Now,
$n\left(M\cap P\right)=30,n\left(M\cap C\right)=28,n\left(P\cap C\right)=23\mathrm{and}n\left(M\cap P\cap C\right)=18$
Number of students who opted for only mathematics:
$$\begin{gathered} n\left(M\cap P\text{'}\cap C\text{'}\right)=n\left\{M\cap \left(P\cup C\right)\text{'}\right\} \\ =n\left(M\right)-n\left\{M\cap \left(P\cup C\right)\right\} \\ =n\left(M\right)-n\left\{\left(M\cap P\right)\cup \left(M\cap C\right)\right\} \\ =n\left(M\right)-\left\{n\left(M\cap P\right)+n\left(M\cap C\right)-n\left(M\cap P\cap C\right)\right\} \\ =100-\left(30+28-18\right) \\ =60 \end{gathered}$$
Therefore, the number of students who opted for mathematics alone is 60