Question

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?

Answer 1

Step 1: Given data.
Number of medals won in football $=n\left(F\right)=38$
Number of medals won in basketball $=n\left(B\right)=15$
Number of medals won in cricket $=n\left(C\right)=20$
Number of medals in either football, basketball or cricket $=n(F\cup B\cup C)=58$
Number of medals won in football, basetball and cricket $=n(F\cap B\cap C)=3$

Step 2: Solve for men who received medals in exactly two of the three sports.
$n(F\cup B\cup C)=n\left(F\right)+n\left(B\right)+n\left(C\right)-n(F\cap B)-n(F\cap C)-n(B\cap C)+n(F\cap B\cap C)$
$\Rightarrow 58=38+15+20-n(F\cap B)-(F\cap C)-n(B\cap C)+3$
$\Rightarrow 58=76-n(F\cap B)-n(F\cap C)-n(B\cap C)$
$\Rightarrow n(F\cap B)+n(F\cap C)+n(B\cap C)=76-58$
$\therefore n(F\cap B)+n(F\cap C)+n(B\cap C)=18\cdots \left(i\right)$

Step $3:$ Solve for $a+b$ from Venn diagram.
Let $a$ denote the number of people who got medal in football and basketball but not cricket.
Let $b$ denote the number people who got medal in football and cricket but not basketball.
Let $c$ denote the number of people who got the medals in cricket and basketball but not in football.
Let $d$ denote the number of people who got medals in football and basketball and cricket.


$\therefore d=n(F\cap B\cap C)=3$
And number of people who get exactly two medals $=a+b+c$
$n(F\cap B)+n(F\cap C)+n(B\cap C)=18\left(\text{From}\right(i\left)\right)$
$\Rightarrow (a+d)+(b+d)+(c+d)=18$
$\Rightarrow a+b+c+3d=18$
$\Rightarrow a+b+c=18-3d$
$\Rightarrow a+b+c=18-3\times 3$
$\Rightarrow a+b+c=18-9$
$\Rightarrow a+b+c=9$
$\therefore 9$ people received medals in exactly two of the three sports.