Question

Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basketball.Of the total ,64 played both basketball and hockey , 80 played cricket and basketball and 40 played cricket and hockey; 24 played all the three games.Find     i)how many played only only cricket,basketball and only cricket,only hockey,only basket ball?    ii)how many played at least one game?   iii)how many played only two games?   iv)how many do not play any of the 3 games?

Answer 1

Let A = set of boys played Cricket,
Let B = set of boys played Hockey &
Let C = set of boys played Basketball.

i).
- How many boys played only cricket & basketball ?
Solution :
n(A intersection C) = 80 

- How many boys played only cricket ?
Solution :
No. of boys played only cricket
= n(A) - n(A intersection B) - n(A intersection C)
= 224 - 40 - 80 
= 104

- How many boys played only hockey ?
Solution :
No. of boys played only hockey
= n(B) - n(B intersection A) - n(B intersection C)
= 240 - 40 - 64 
= 136

- How many boys played only basketball ?
Solution :
No. of boys played only basketball
= n(C) - n(C intersection A) - n(C intersection B)
= 336 - 80 - 64 
= 192

ii).How many played the game ? 
Solution :
= No. of boys participated 
= n(A U B U C)
= n(A) + n(B) + n(C) - n(A intersection B) - n(B intersection C) 
 - n(C intersection A) + n(A intersection B intersection C)
= 224 + 240 + 336 - 40 - 64 - 80 + 24
= 640

iii).How many played only two games ? 
Solution :
No. of boys played only two games 
= No. of boys participated - No. of boys played only cricket - No. of boys played only hockey - No. of boys played only basketball - No. of boys played all games
= 640 - 104 - 136 - 192 - 24
= 184

iv).How many do not play any of the 3 games ?
Solution :
No. of boys who do not played any game
= Total No. of boys -  No. of boys participated
= 800 - 640 
= 160