Question

Out of $60$ students in a class, anyone who has chosen to study maths elects to do physics as well. But no one does maths and chemistry, $16$ do physics and chemistry. All the students do at least one of the three subjects and the number of people who do exactly one of the three is more than the number who do more than one of the three. Then the range of cardinal number of students who could have done only chemistry is

• A. $[0,40]$
• B. $[0,44]$
• C. $[2,28]$
• D. $[2,38]$

Answer 1

The correct option is B $[0,44]$

Students who do math also do physics.
$\Rightarrow M\subset P$


Let $n(M\cap P)=\alpha ,$
Students who elect only $P$ be $\beta$
and Students who elect only $C$ be $\gamma .$
Then, $\alpha +\beta +\gamma +16=60$
$\Rightarrow \alpha +\beta +\gamma =44$
Also, $\beta +\gamma >\alpha +16$

Now, possibilities of $\gamma :$
$\left(i\right)$ If $\gamma =0,$ then $\alpha +\beta =44$
And $\beta >\alpha +16$
Here, $(\alpha ,\beta )=(1,43),(2,42),(3,41),\dots ,(13,31)$

$\left(ii\right)$ If $\gamma =44,$ then $\alpha +\beta =0$
$\Rightarrow \alpha =\beta =0,$ this is also possible.
$\therefore$ Range of $\gamma$ is $[0,44]$