How many play both the games?

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Solution

Let A and B be the sets of those playing hockey and cricket, respectively.
Then $n (A) = 28$ , $n (B) = 33$ and $n (A \cup B) = 60 - 14 = 46$ .
Number of pupils who play both $= n (A \cap B) = n (A) + n (B) - n (A \cup B) = 28 + 33 - 46 = 15$
Note: The formulae used above can be extended for more than two sets also.
For any three sets A, B, and C,
$n (A \cup B \cup C) = n (A) + n (B) + n (C) - n (A \cap C) - n (B \cap C) - n (A \cap C) + n (A \cap B \cap C)$

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