Question

In the class of $70$ students, $45$ students like to play cricket, $52$ students like to play kho-kho and all the students like to play at least one of the two games. Represent this information using the Venn diagram. How many students like to play both the games cricket and kho-kho?

Answer 1

Hint: Let the set of students playing cricket be $A$ and the set of them playing kho-kho be $B$ and it is given that $n(A) = 45,n(B) = 52$ and we know that $n(A \cup B)$ is the set of students who like to play at least one of the two games. So $n(A \cup B) = 70$ and $n(A \cap B)$ are those who like to play both the games and we know the formula that
$n(A \cup B)$$ = n(A) + n(B) - $$n(A \cap B)$

Complete step-by-step answer:
Here in the above question, we are given that in the class of $70$ students, $45$ students like to play cricket, $52$ students like to play kho-kho and all the students like to play at least one of the two games.
Let the set of students playing cricket be $A$ and the set of them playing kho-kho be $B$ and it is given that $n(A) = 45,n(B) = 52$
So set $A = ${students who like to play cricket}
Set $B = ${students who like to play kho-kho}
Hence $n(A) = 45,n(B) = 52$
We know that $n(A \cup B)$ is the set of students who like to play at least one of the two games. So $n(A \cup B) = 70$ and $n(A \cap B)$ are those who like to play both the games and we know the formula that
$n(A \cup B)$$ = n(A) + n(B) - $$n(A \cap B)$
               

                                   AႶB
Hence we know the formula that
$n(A \cup B)$$ = n(A) + n(B) - $$n(A \cap B)$
$n(A)$ represents number of students liking cricket
$n(B)$ represents number of students who like kho-kho
$n(A \cup B)$ are the number of students liking at least one game
$n(A \cap B)$ are those who like both the games.
$70$$ = 45 + 52 - $$n(A \cap B)$
Hence we get that
$n(A \cap B) = 97 - 70 = 27$
So those who like both the games are $27$ students.

Note: $n(A \cup B)$ is called as $A{\text{ union }}B$ which means that include both
$n(A \cap B)$ is called as $A{\text{ intersection }}B$ which means that it is their common point.
So it is given that
$n(A \cup B)$$ = n(A) + n(B) - $$n(A \cap B)$