Question

There are $100$ students in a class. It is found that $60$ students study Mathematics, $45$ students study Physics and $35$ students study Chemistry. Moreover, $20$ students study Mathematics and Physics, $15$ students study Physics and Chemistry, $25$ students study Chemistry and Mathematics and $10$ students study none of these subjects. Then

• A. $95$ students study at least one of Mathematics, Physics and Chemistry
• B. $10$ students study each of Mathematics, Physics and Chemistry
• C. $70$ students study either Mathematics or Chemistry
• D. $35$ students study Mathematics along with either Physics or Chemistry

Answer 1

Answer: (B), (C), (D)

The correct options are
B $10$ students study each of Mathematics, Physics and Chemistry
C $35$ students study Mathematics along with either Physics or Chemistry
D $70$ students study either Mathematics or Chemistry

Given there are $100$ students in a class,

let $M$ be the set of people who study mathematics,$\therefore n\left(M\right)=60$,

let $P$ be the set of people who study physics, $\therefore n\left(P\right)=45$,

let $C$ be the set of people who study chemistry, $\therefore n\left(C\right)=35$,

Also given that $n(M\cap P)=20$, $n(P\cap C)=15$ and $n(M\cap C)=25$

$10$ students study none of these subjects

$\therefore 100-n(M\cup P\cup C)=10$

$\Rightarrow n(M\cup P\cup C)=90$

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

$\Rightarrow 90=60+45+35-20-15-25+n(M\cap P\cap C)$

$\Rightarrow n(M\cap P\cap C)=10$

So, option B is correct.

Now, $n(M\cup C)=n\left(M\right)+n\left(C\right)-n(M\cap C)=60+35-25=70$

So, option C is correct.

Also, $n(P\cup C)=n\left(P\right)+n\left(C\right)-n(P\cap C)=45+35-15=65$

To find $n(M\cap (P\cup C\left)\right)$

$n(M\cap (P\cup C\left)\right)=n\left(M\right)+n(P\cup C)-n(M\cup P\cup C)=60+65-90=35$

Hence, option B,C and D are correct.