In a group of 24 children, each one plays cricket or hockey or both. If 16 play cricket and 12 play cricket only, find how many play hockey only.

Given:

Total number of children: n(C ∪ H) = 24

Children who play cricket : n(C) = 16

Children who play cricket only: n(C - H) = 12

Children who play hockey only: n(W - C) = ?

First find how many children play both using the formula:

n(C ∩ H) = n(C) - n(C - H)

Substituting the values in above, we get:

n(C ∩ H) = 16 - 12

n(C ∩ H) = 4

So, 4 children play both cricket and hockey.

Since every child in the group of 24 plays at least one game, the total is the sum of “cricket only,” “hockey only,” and “both.”

n(C ∪ H) = n(C - H) + n(H - C) + n(C ∩ H)

n(H - C) = n(C ∪ H) - n(C - H) - n(C ∩ H) $\quad$[Solving for n(H - C)]

Substituting the values in above, we get:

n(H - C) = 24 - 12 - 4

n(H - C) = 24 - 16

n(H - C) = 8

∴ Number of children who play hockey only = 8.

The Venn diagram is shown below: