Question

If 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 $18$. The number of students who have opted Mathematics alone is

• A. $35$
• B. $48$
• C. $60$
• D. $22$

Answer 1

Answer: (C)

$60$

Answer 2

Let $M, P$ and $C$ be the set of students who opt Mathematics, Physics and Chemistry respectively.
Then $n(M) = 100, n(P) = 70, n(C) = 40$
$n(M \cap P) = 30, n(M \cap C) = 28, n(P \cap C) = 23,$
$n(M \cap P \cap C) = 18$
Number of students who opted Mathematics
alone = $n(M \cap P' \cap C')$
= $n(M \cap (P \cup C)')$
= $n(M) - n(M \cap (P \cup C))$
= $n(M) - n((M \cap P) \cup (M \cap C))$

= $n(M)- n(M \cap P)- n(M \cap C) + n(M \cap P \cap C)$
= $100 - 30 - 28 + 18 = 60$