Question

In a class there are $200$ students, at least $140$ of students like Maths, at least $150$ like Science and at least $160$ like English. All students like atleast $1$ subject. What is the minimum number of students who like all three subjects?

• A. 50
• B. 83
• C. 100
• D. 150

Answer 1

The correct option is A

50

We know that,
$a+d+g+f\ge 140$
$b+d+g+e\ge 150$
$c+e+f+g\ge 160$

Adding these inequalities we get,
$(a+b+c)+2(d+e+f)+3g\ge 450\cdots \left(i\right)$

We know that,
$a+b+c+d+e+f+g=200\cdots \left(ii\right)$

Multiplying $\left(ii\right)\text{with}2$ and subtracting it from $\left(i\right)$ we get,

$g-(a+b+c)\ge 50$
$\Rightarrow g\ge 50+(a+b+c)$

$(a+b+c)$ is the number of students who like $1$ subject alone.

for $g$ to be minimum, $(a+b+c)=0$
and hence,

$g\ge 50$

So a minimum of $50$ students like all $3$ subjects