A college awarded 38 medals in football, 15 in basketball and 20 in cricket if these medals went to total of 58 men and only three men got medals in all three sports. Number of person who get medals in exactly two of the three sports is:

15

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9

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18

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12

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Solution

The correct option is D $9$
$\begin{matrix} n (F \cup B \cup C) = n (F) + n (B) + n (C) - n (B \cap C) - n (F \cap B) - n (F \cap C) + n (F \cup B \cup C) 58 = 38 + 15 + 20 - [n (B \cap C) + n (F \cap B) + n (F \cap C)] 18 = [n (B \cap C) + n (F \cap B) + n (F \cap C)] \end{matrix}$

Total number of men received medals in two sports
$\begin{matrix} = n (F \cap B \cap C^{'}) + n (F \cap B^{'} \cap C) + n (F^{'} \cap B \cap C) = n (B \cap C) + n (F \cap B) + n (F \cap C) - 3 n (F \cap B \cap C) = 9 \end{matrix}$
Because, ( $n (F \cap B \cap C^{'}) = n (F \cap B) - n (F \cap B \cap C)$ )

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