Question

In a class of 60 students, 25 students play cricket, 20 students play tennis and 10 students play both the games, then the number of students who play neither are
(a). 0
(b). 35
(c). 45
(d). 25

Answer 1

Hint: Find the number of students who play both cricket and tennis by finding $n\left( A\cup B \right)$. From this the students who play neither cricket nor tennis can be formed by subtracting from total students.

Complete step-by-step answer:
Given the total number of students in a class = 60.
Let ‘A’ be the set of students who play cricket, which is 25 in number.
$\therefore n\left( A \right)=25$
Let ‘B’ be the set of students who play tennis, 20 in number.
$\therefore n\left( B \right)=20$
The number of students who play both cricket and tennis is 10.
$\therefore n\left( A\cap B \right)=10$

The shaded area shows $A\cap B$.
The intersection of two sets A and B, consist of all elements that are both in A and B. The figure shows a Venn diagram representing the same.
Here, we are asked to find the number of students who don’t play cricket or tennis. Thus we need to find $\left( A\cup B \right)$ and subtract it from the total number of students.
$A\cup B$ is A union B, which means creating a new set containing every element from either of A and B.
The given Venn diagram represents $A\cup B$.

Hence, $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$
This formula can be directly derived from the above Venn diagram,
$\therefore n\left( A\cup B \right)=25+20-10=35$.
Here, 35 students play at least one out of cricket or tennis out of 60 students in a class.
$\therefore$The number of students who play neither cricket nor tennis =
Total students – number of students who play at least one game
= Total students - $n\left( A\cup B \right)$
= 60 – 35 = 25
$\therefore$The number of students who play neither cricket nor tennis = 25.
Hence, option (d) is the correct answer.

Note: A Venn diagram is used to represent all possible relations of different sets. Here we used $A\cap B$, which is the intersection of 2 sets to represent the common elements in both set A and B. And $A\cup B$represents the combined elements of set A and B.
Care should be taken not to confuse between $A\cap B$ and $A\cup B$.