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 were offered all the three subjects, find how many took Mathematics but not Chemistry.

Answer 1

Find the required number of students.

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 the three subjects.

So, the given data can be represented as follows:

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

The total number of students who took Mathematics but not Chemistry can be given as:

$$\begin{gathered} n\left(M\right)-n(C\cap M)=24-13 \\ \Rightarrow n\left(M\right)-n(C\cap M)=11 \end{gathered}$$

Therefore, the total number of students who took Mathematics but not Chemistry is $11$.