Question

Each student in a class of 40 plays at least one indoor game: chess, backgammon, or scrabble. 18 play chess, 20 play scrabble and 27 play backgammon. 7 play chess and scrabble, 12 play scrabble and backgammon. Lastly, 4 play chess, backgammon and scrabble. Find the number of students who play chess and backgammon. Find the number of students who play chess, backgammon but not scrabble.

Answer 1

Answer:-

$\pink{\bigstar}$ The number of students who play chess and backgammon are 10. 

$\pink{\bigstar}$ The number of students who play chess, backgammon and not scrabble are 6.

• Given:-

Total number of students = 40

Students playing chess = 18

Students playing scrabble = 20

Students playing backgammon = 27

• Solution:-

Let A be the set of students who play chess 

Let B be the set of students who play scrabble 

Let C be the set of students who play backgammon 

• ATQ:-

n(A ∪ B ∪ C) = 40

n(A) = 18

n(B) = 20

n(C) = 27

n(A ∩ B) = 7

n(C ∩ B) = 12

n(A ∩ B ∩ C) = 4 

We have :- 

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(C ∩ A) + n(A ∩ B ∩ C) 

→ 40 = 18 + 20 + 27 - 7 - 12 - n(C ∩ A) + 4 

→ 40 = 69 - 19 - n(C ∩ A) 

→ 40 = 50 - n(C ∩ A)

→ n(C ∩ A) = 50 - 40 

→ n(C ∩ A) = 10 

Therefore, the number of students who play chess and backgammon are 10. 

The number of students who play chess, backgammon and not scrabble are:- 

→ n(C ∩ A) - n(A ∩ B ∩ C) 

→ 10 – 4 

→ 6

Therefore,the number of students who play chess, backgammon and not scrabble are 6.