Question

In a group of 100 people, 65 like to play cricket, 40 like to play
tennis and 55 like to play volleyball. All of them like to play
at least one of three games. if 25 like to play both cricket and
tennis, 24 like to play both tennis and volleyball and 22 like to
play both cricket and volleyball then

How many like to play all the three games?
How many like to play cricket only?
How many like to play tennis only?​

Answer 1

Given:

Number of people who like to play cricket, $n(C)$ = 65

Number of people who like to play tennis, $n(T)$ = 40

Number of people who like to play volleyball, $n(V)$ = 55

Total number of people, $n(C\cap T \cap V)=100$

Number of people who play cricket and tennis, $n(C \cap T)$ = 25

Number of people who play tennis and volleyball, $n(T \cap V)$ = 24

Number of people who play cricket and volleyball, $n(C \cap V)$ = 22

To find:

1. Number of people who like to play all three games.

2. Number of people who like to play cricket only.

3. Number of people who like to play tennis only.

Solution:

Formula:

$n(A\cup B \cup C) = n(A)+n(B) +n(C)-n(A \cap B)-n(B \cap C)-n(A \cap C)+n(A\cap B\cap C)$

So, we can write the formula as per our question as:

$n(C\cup T \cup V) = n(C)+n(T) +n(V)-n(C \cap T)-n(T \cap V)-n(C \cap V)+n(C\cap T\cap V)$

Putting the values:

$100 = 65+40+55-25-24-22+n(C\cap T\cap V)\\\Rightarrow n(C\cap T\cap V) = 171-65-55-40\\\Rightarrow \bold{n(C\cap T\cap V) = 11}$

Answer 1: 11

2. Number of people who like to play cricket only:

$n(C\ only)=n(C)-n(C\cap T)-n(C\cap V)+n(C\cap T\cap V)$

$n(C\ only)=65-25-22+11 = \bold{29}$

Answer 2: 29

3. Number of people who like to play tennis only:

$n(T\ only)=n(T)-n(T\cap C)-n(T\cap V)+n(C\cap T\cap V)$

$n(T\ only)=40-25-24+11 = \bold{2}$

Answer 3: 2