Question

From $50$ students taking examinations in Mathematics, Physics and Chemistry, $37$ passed Mathematics, $24$ Physics and $43$ Chemistry. At most $19$ passed Mathematics and Physics, at most $29$ passed Mathematics and Chemistry and at most $20$ passed Physics and Chemistry. The largest possible number that could have passed all three examinations is

• A. $11$
• B. $12$
• C. $13$
• D. $14$

Answer 1

Answer: (D)

Let $M$, $P$ and $C$ be the sets of students taking examinations in Mathematics, Physics and Chemistry respectively.
$\therefore n\left(M\cup P\cup C\right)=50$,
$n\left(M\right)=37$, $n\left(P\right)=24$, $n\left(C\right)=43$,
$n\left(M\cap P\right)\le 19$, $n\left(M\cap C\right)\le 29$, $n\left(P\cap C\right)\le 20$.
We have
$n\left(M\cup P\cup C\right)=n\left(M\right)+n\left(P\right)+n\left(C\right)-n\left(M\cap P\right)$
$-n\left(M\cap C\right)-n\left(P\cap C\right)+n\left(M\cap P\cap C\right)$
$\Rightarrow 50=37+24+43-n\left(M\cap P\right)-n\left(M\cap C\right)-$
$n\left(P\cap C\right)+n\left(M\cap P\cap C\right)$
$\Rightarrow n\left(M\cap P\cap C\right)=n\left(M\cap P\right)+n\left(M\cap C\right)+n\left(P\cap C\right)-54$
$\Rightarrow n\left(M\cap P\cap C\right)\le 19+29+20-54=14$
$\Rightarrow n\left(M\cap P\cap C\right)\le 14$.