Question

In a class $18$ students took Physics, $23$ students took Chemistry and $24$ students took Mathematics of those $13$ took both Chemistry and Mathematics, $12$ took both Physics and Chemistry and $11$ took both Physics and Mathematics. If $6$ students offered all the three subjects, find out how many took exactly one of the three subjects.

Answer 1

Find the number of students who took only one subject.

Assume that, $P$ denotes Physics, $C$ denotes Chemistry, $M$ denotes Mathematics and $n$ denotes the number of students.

So, $n\left(C\cap M\right)$ denotes the number of students who took both Chemistry and Mathematics.

$n\left(P\cap C\right)$ denotes the number of students who took both Physics and Chemistry.

$n\left(P\cap M\right)$ denotes the number of students who took both Physics and Mathematics.

And $n\left(P\cap C\cap M\right)$ denotes the number of students who took all three subjects.

So, the given data can be represented as follows:

$$\begin{gathered} n\left(P\right)=18 \\ n\left(C\right)=23 \\ n\left(M\right)=24 \\ n(C\cap M)=13 \\ n(P\cap C)=12 \\ n(P\cap M)=11 \\ n(P\cap C\cap M)=6 \end{gathered}$$

When we add the number of students who took Physics and the number of students who took Chemistry and the number of students who took mathematics, the number of students who took both Chemistry and Mathematics and the number of students who took both Physics and Chemistry and the number of students who took both Physics and Mathematics are counted two times and the number of students who took all the three subjects are counted three times.

So, to find out how many took exactly one of the three subjects we have to subtract twice the number of students who took two subjects and three subjects out of the sum of the number of students who took Physics, chemistry and mathematics.

The total number of students who took only one subject can be given as:

$$\begin{gathered} n\left(P\right)+n\left(C\right)+n\left(M\right)-2n(P\cap C)-2n(C\cap M)-2n(P\cap M)+3n(P\cap C\cap M)=18+23+24-2·13-2·12-2·11+3·6 \\ \Rightarrow n\left(P\right)+n\left(C\right)+n\left(M\right)-2n(P\cap C)-2n(C\cap M)-2n(P\cap M)+3n(P\cap C\cap M)=11 \end{gathered}$$

Therefore, the total number of students who took only one subject is $11$.