Question

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

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

Answer 1

Answer: (D)

$14$

Let M, P and C be the sets of students taking examinations in Mathematics, Physics and Chemistry, respectively.
We have,

$n(M\cup P\cup C)=50,n\left(M\right)=37,n\left(P\right)=24$,
$n\left(C\right)=43$
$n(M\cap P)\le 19,n(M\cap C)\le 29,n(P\cap C)\le 20$

Now, $n(M\cup P\cup C)=n\left(M\right)+n\left(P\right)+n\left(C\right)-n(M\cap P)-n(M\cap C)-n(P\cap C)+n(M\cap P\cap C)$

$\Rightarrow 50=37+24+43-\left\{n\right(M\cap P)+n(M\cap C)+n(P\cap C\left)\right\}+n(M\cap P\cap C)$

$\Rightarrow n(M\cap P)+n(M\cap C)+n(P\cap C)=n(M\cap P\cap C)+54$ .......(i)

Now, $n(M\cap P)+n(M\cap C)+n(P\cap C)\le 19+29+20$ [using Eq. (i)]
$\Rightarrow n(M\cap P\cap C)+54\le 68$

$\Rightarrow n(M\cap P\cap C)\le 14$.