A college awarded 38 medals in football,15 in | vedclass

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(B) Let $F, B,$ and $C$ denote the set of men who received medals in football,basketball,and cricket,respectively.Given:$n(F) = 38, n(B) = 15, n(C) =

Vedclass · Accepted Answer

$9$

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(B) Let $F, B,$ and $C$ denote the set of men who received medals in football,basketball,and cricket,respectively.Given:$n(F) = 38, n(B) = 15, n(C) = 20$$n(F \cup B \cup C) = 58$$n(F \cap B \cap C) = 3$Using the formula:$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)$Substituting the values:$58 = 38 + 15 + 20 - [n(F \cap B) + n(F \cap C) + n(B \cap C)] + 3$$58 = 76 - [n(F \cap B) + n(F \cap C) + n(B \cap C)]$$n(F \cap B) + n(F \cap C) + n(B \cap C) = 76 - 58 = 18$Let $a, b, c$ be the number of men who received medals in exactly two sports,and $d$ be the number of men who received medals in all three sports.$n(F \cap B) = a + d$$n(F \cap C) = b + d$$n(B \cap C) = c + d$Summing these:$(a + d) + (b + d) + (c + d) = 18$$a + b + c + 3d = 18$Since $d = 3$:$a + b + c + 3(3) = 18$$a + b + c + 9 = 18$$a + b + c = 9$Thus,$9$ men received medals in exactly two of the three sports.

$A$ college awarded $38$ medals in football,$15$ in basketball and $20$ in cricket. If these medals went to a total of $58$ men and only three men got medals in all the three sports,how many received medals in exactly two of the three sports?

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