In a group of children, 45 play football out of which 30 play football only 28 play hockey, 25 play cricket, out of which 11 play cricket only, further, 7 play cricket and football but not hockey, 5 play football and hockey but not cricket and 10 play football and cricket both. Find:

a) How many play all 3 games?
b) How many children play hockey only?
c) How many children are there in the group?

SOLUTION

To solve this problem, we can use the principle of inclusion-exclusion.

a) To find how many children play all 3 games, we need to find the intersection of the sets representing each game. Let's define the sets:
- A: Set of children who play football.
- B: Set of children who play hockey.
- C: Set of children who play cricket.

We know that the number of children who play football only (excluding hockey and cricket) is 30. Similarly, the number of children who play cricket only is 11. The number of children who play football and hockey but not cricket is 5. And the number of children who play cricket and football but not hockey is 10.

Using these values, we can calculate the number of children who play all 3 games by using the formula for the intersection of three sets:
|A ∩ B ∩ C| = |A ∪ B ∪ C| - (|A| + |B| + |C|) + (|A ∩ B| + |A ∩ C| + |B ∩ C|) - |A ∩ B ∩ C|

Substituting the values into the formula:
|A ∩ B ∩ C| = 45 - (30 + 28 + 25) + (7 + 10 + 5) - |A ∩ B ∩ C|

Simplifying the expression, we have:
|A ∩ B ∩ C| = 45 - 83 + 22 - |A ∩ B ∩ C|
|A ∩ B ∩ C| = -16

Since the number of children cannot be negative, there is a mistake in the given data or question. Please double-check the information provided.

b) To find how many children play hockey only, we can calculate the size of the set B - (A ∪ C). This set represents the children who play hockey but not football or cricket.

Size of B - (A ∪ C) = |B - (A ∪ C)| = |B| - |A ∩ B| - |B ∩ C| + |A ∩ B ∩ C|

Substituting the values into the formula:
|B - (A ∪ C)| = 28 - 5 - 7 + |A ∩ B ∩ C|
|B - (A ∪ C)| = 16

So, there are 16 children who play hockey only.

c) To find the total number of children in the group, we can calculate the size of the union of the three sets A, B, and C.

Size of A ∪ B ∪ C = |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

Substituting the values into the formula:
|A ∪ B ∪ C| = 45 + 28 + 25 - 7 - 10 - 5 + |A ∩ B ∩ C|
|A ∪ B ∪ C| = 76

So, there are 76 children in the group.

(a) 3

(b) 16
(c) 76

how many play all three games.

AnsAnswer