A college awarded 38 medals in football, 15 in basketball and 20 in cricket. If the medals went to a total of 58 men and only three men got medals in all the three sports, then the number of men received medals in exactly two of the three sports is

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answer is 9.

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Detailed Solution

$n (F) = 38, n (B) = 15, n (C) = 20$

$n (F \cup B \cup C) = 58, n (F \cap B \cap C) = 3$

$n (F \cup B \cup C) = n (F + n (B) + n (C) - n (F \cap B) - n (F \cap C) - n (B \cap C) + n (F \cap B \cap C))$

$\Rightarrow n (F \cap B) + n (F \cap C) + n (B \cap C) = 18$

a, denote number of men who got medals in foot ball and basket ball.
b, denote number of men who got medals foot ball and cricket.
c, denote number of men who got medals basket ball and cricket.
d, denote number of men who got medals in all the three sports
$∴ a + d + b + d + c + d = 18$ (d = 3) or $a + b + c = 9$