Question

In a class of 55 students, the numbers of students studying in different subjects are, 23 in Mathematics, 24 in Physics, 19 in Chemistry, 12 in Mathematics and Physics, 9 in Mathematic 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

Hint: Here, we will first draw the Venn diagram and then we will find the number of students studying only Mathematics, only Physics and only Chemistry one by one using the Venn diagram. At last, the sum of the number of students who are studying only Mathematics or Physics or Chemistry will give us the total number of students who are studying exactly one subject.

Complete step-by-step answer:
The Venn diagram for the given case can be drawn as:

Here, region 1 represents $ M\cap P $ which means students studying Mathematics and Physics.
Region 2 represents $ M\cap C $ which means students studying Mathematics and Chemistry.
Region 3 represents $ P\cap C $ which means students studying Physics and Chemistry.
Region 4 represents $ M\cap P\cap C $ which means students studying Mathematics, Physics as well as Chemistry.
First of all we will find the number of students studying only mathematics, which is given as:
 $ n\left( M \right)-n\left( M\cap P \right)-n\left( M\cap C \right)+n\left( M\cap P\cap C \right)..........\left( 1 \right) $
Here, $ n\left( M \right) $ = total number of students studying mathematics = 23
 $ n\left( M\cap P \right) $ = number of students studying Mathematics and Physics = 12
 $ n\left( M\cap C \right) $ = number of students studying Mathematics and Chemistry = 9
 $ n\left( M\cap P\cap C \right) $ = number of students studying Mathematics, Physics as well as Chemistry = 4
We have to add \[n\left( M\cap P\cap C \right)\] once because \[\left( M\cap P\cap C \right)\] is contained in \[\left( M\cap P \right)\] as well as in \[\left( M\cap C \right)\] and thus it gets subtracted twice.
On putting the respective values in equation (1), we get:
Number of students studying only Mathematics = 23 - 12 – 9 + 4 =6
Similarly, number of students studying only Chemistry is given as:
 $ \begin{align}
  & =n\left( C \right)-n\left( C\cap M \right)-n\left( C\cap P \right)+n\left( M\cap P\cap C \right) \\
 & =19-9-7+4=7 \\
\end{align} $
And, the number of students studying only Physics is given as:
 $ \begin{align}
  & =n\left( P \right)-n\left( P\cap C \right)-n\left( P\cap M \right)+n\left( M\cap P\cap C \right) \\
 & =24-7-12+4=9 \\
\end{align} $
So, the number of students who study exactly one subject is = 6+7+9= 22.
Hence, 22 students study only one subject.

Note: Students should note here that we have to add $ n\left( P\cap C\cap M \right) $ in all cases because in all the cases both the regions that are subtracted contain $ n\left( P\cap C\cap M \right) $ , so if we do not add it, we will get wrong result . All the calculations must be done properly to avoid mistakes.