Question

There are $100$ students in a class. In an examination, $50$ of them failed in Mathematics, $45$ failed in Physics, $40$ failed in Biology and $32$ failed in exactly two of the three subjects. Only one student passed in all the subjects. Then, the number of students failing in all the three subjects is

• A. $12$
• B. $4$
• C. $2$
• D. Cannot be determined from the given information

Answer 1

The correct option is C $2$

Given that there are $100$ students and only one student passed in all $3$ subjects

So the number of students failed in at least one subject is $99$

Let $a$ be the number of students failed only in mathematics, $b$ be number of students failed only in physics and $c$ be number of students failed only in Biology.

Let $x$ be number of students failed only in mathematics and physics, $y$ be number of students failed only in physics and biology and $z$ be the number of students failed only in biology and mathematics.

Let $p$ be the number of students failed in all three subjects

By Venn diagram, we get $99=a+b+c+x+y+z+p$

Number of students failed in exactly two subjects is $32=x+y+z$

Students failed in mathematics is $50=a+x+z+p$ , students failed in physics is $45=b+x+y+p$ and students failed in Biology is $40=c+y+z+p$

By adding above three equations, we get $135=a+b+c+2(x+y+z)+3p$

$\Rightarrow 135=99+32+2p$

$\Rightarrow p=2$

Therefore option $C$ is correct