Question

In a class of $55$ students, the number of students studying in different subject 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 three subjects. Find the number of students who have taken exactly one subject.

Answer 1

Given , $N=55,n\left(M\right)=23,n\left(P\right)=24,n\left(C\right)=19,n(M\cap P)=12,n(P\cap C)=7,n(M\cap C)=9,n(M\cap P\cap C)=4$
Now, number of students studying only Mathematics
$n(M\cap P^{\prime }\cap C^{\prime })=n\left(M\right)-n(M\cap P)-n(M\cap C)+n(M\cap P\cap C)$ (by Venn diagram)
$=23-9-12+4=6$
Now, number of students studying only Physics
$n(P\cap M^{\prime }\cap C^{\prime })=n\left(P\right)-n(P\cap M)-n(P\cap C)+n(P\cap M\cap C)$ (by Venn diagram)
$=24-12-7+4=9$
Now, number of students studying only Chemistry
$n(C\cap M^{\prime }\cap P^{\prime })=n\left(C\right)-n(C\cap M)-n(C\cap P)+n(M\cap P\cap C)$ (by Venn diagram)
$=19-9-7+4=7$
So, the number of people who study exactly one of the three subjects $=6+9+7=22$