Each student in a clas of 40 plays at least one indoor game chess, carrom and scrabble 18 play - Maths - Sets

Question

Each student in a clas of 40 plays at least one indoor game chess,
carrom and scrabble 18 play chess, 20 play scrabble and 27 play
carrom 7 play chess and scrabble, 12 play scrabble and carrom and 4
play chess, carrom and scrabble Find the number of students who
play (i) chess and scrabble (ii) chess, carrom but not scrabble

Alan Poulose · Answer

Solution

Let A be the set of students who play chess

B be the set of students who play scrabble

C be the set of students who play carrom

Therefore, We are given 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)

Therefore, 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, Number of students who play chess and carrom are 10

Also, number of students who play chess, carrom and not
scrabble

= n(C ∩ A) - n(A ∩ B ∩ C)

= 10 – 4

= 6