Question

In a class of $35$ students, $17$ have Chemistry, $10$ have Chemistry but not Physics. If each student of the class has taken either Chemistry or Physics or both, then which of the following is/are true?

• A. Number of students taking both Chemistry and Physics is $7$.
• B. Number of students taking both Chemistry and Physics is $9$.
• C. Number of students taking Physics but not Chemistry is $18$.
• D. Number of students taking Physics but not Chemistry is $20$.

Answer 1

Answer: (A), (C)

Number of students taking Physics but not Chemistry is $18$.

Let $C$ denote the set of students who have taken Chemistry and $P$ be the set of students taking Physics.
It is given that
$n(C\cup P)=35,n\left(C\right)=17,\text{and}n(C-P)=10.$
Now, $n(C-P)=n\left(C\right)-n(C\cap P)$
$\Rightarrow 10=17-n(C\cap P)$
$\Rightarrow n(C\cap P)=7$

Number of students having both Chemistry and Physics are $7$.
$n(C\cup P)=n\left(C\right)+n\left(P\right)-n(C\cap P)$
$\Rightarrow 35=17+n\left(P\right)-7$
$\Rightarrow n\left(P\right)=25$

$n(P-C)=n\left(P\right)-n(C\cap P)$
$\Rightarrow \left(n\right(P-C)=25-7$
$\Rightarrow \left(n\right(P-C)=18$

Tne number of students taking Physics but not Chemistry is $18$.