In a class of $55$ students, the number of students studying different subjects are $23$ in Mathematics, $24$ in Physics, $19$ in Chemistry, $12$ in Mathematics and Physics, $9$ in Mathematics and Chemistry, $7$ in Physics and Chemistry and $4$ in all the subjects. The number of students who have taken exactly one subject is

The correct option is B $22$

Now we have given total student as : $55$

So let $n\left(M\right)$ = student studying maths

$n\left(C\right)$=student studying chemistry

$n\left(P\right)$=student studying Physics

So:

$n\left(M\right)=23,n\left(P\right)=24,n\left(C\right)=19,n(M\cap P)=12,n(M\cap C)=9$ ,

$n(P\cap C)=7,n(M\cap P\cap C)=4$

So now, number of student who study maths but not physics and chemistry are as follows:

$n\left(M\right)-\left[\right(n(M\cap C)+n(M\cap P)]+n(M\cap P\cap C)$

$⟹23-[9+12]+4=6$

Now, student study chemistry but not physics and maths are as follows:

$n\left(C\right)-\left[\right(n(M\cap C)+n(P\cap C)]+n(M\cap P\cap C)$

$⟹19-[9+7]+4=7$

At last, student studying physics but not maths and chemistry are as follows:

$n\left(P\right)-\left[\right(n(M\cap P)+n(P\cap C)]+n(M\cap P\cap C)$

$⟹24-[12+7]+4=9$

So the students studying exact one subject are:

$6+7+9=22$