In a group of children, 35 play football out of which 20 play football only, 22 play hockey, 25 play cricket out of which 11 play cricket only. Out of these 7 play cricket and football but not hockey, 3 play football and hockey but not cricket and 12 play football and cricket both.
How many play all the three games ? How many play cricket and hockey but not football, how many play hockey only? What is the total number of children in the group?

Answer

5, 2, 12, 60

Explanation

Given, a = 20 …(i)
e + f + g = 15 …..(ii)
b + d + f + g = 22 …(iii)
c = 11 …(iv)

`c+d+e+g=25 ...(v)`
implies d + e + g = 14 …(vi)
e = 7 …(vii)
f = 3 …(viii)
From Eqs. (vii), (viii) and (ix).
e + g = 12 ...(ix)
e = 7, f = 3, g = 5
From Eq. (vi), d = 2
From Eq. (iii) b + 2 + 3 + 5 = 22
`therefore` b = 12
Hence, a = 20, b = 12, c = 11, d = 2, e = 7, f = 3, g = 5
Number of children play all the three games = g = 5
Number of children play cricket and hockey but not football
= d = 2
Number of children play hockey only = b = 12
Total number of children in the group
`=a+b+c+d+e+f+g=60`