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?
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?
Let $F, B$ and $C$ denote the set of men who received medals in football, basketball and cricket. respectively.
Then $n( F )=38, n( B )=15, n( C )=20$
$n( F \cup B \cup C )=58$ and $n( F \cap B \cap C )=3$
Therefore, $\quad n( F \cup B \cup C )=n( F )+n( B )$
$+n( C )-n( F \cap B )-n( F \cap C )-n( B \cap C )+$
$n( F \cap B \cap C )$
gives $n( F \cap B )+n( F \cap C )+n( B \cap C )=18$
Consider the Venn diagram as given in Fig
Here, $a$ denotes the number of men who got medals in football and basketball only, $b$ denotes the number of men who got medals in football and cricket only, $c$ denotes the number of men who got medals in basket ball and cricket only and $d$ denotes the number of men who got medal in all the three.
Thus, $d=n( F \cap B \cap C )=3$ and $a+d+b+d+c+d=18$
Therefore $a+b+c=9,$
which is the number of people who got medals in exactly two of the three sports.
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