Question

A school awards $77$ medals in three sports i.e. $48$ in football, $25$ in tennis and $25$ in cricket. If $7$ students got medals in all the three sports, then the number of students who received medals in exactly 2 sports is

Answer 1

Let $A$, $B$ and $C$ be the sets of students who won medals in football, tennis and cricket respectively.


Then, $n\left(A\right)=48,n\left(B\right)=25,n\left(C\right)=25,n(A\cap B\cap C)=7$
and $n(A\cup B\cup C)=77$

We know that,
$n(A\cup B\cup C)=n\left(A\right)+n\left(B\right)+n\left(C\right)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C)$

$\Rightarrow n(A\cap B)+n(B\cap C)+n(C\cap A)=n\left(A\right)+n\left(B\right)+n\left(C\right)+n(A\cap B\cap C)-n(A\cup B\cup C)$
$=\left\{\right(48+25+25+7)-77\}=(105-77)=28$

Let $a,b\text{and}c$ be the number of students who won medals in football and tennis both, tennis and cricket both, football and cricket both respectively and $d$ be the number of students who won medals in all three games.

Then,
$n(A\cap B)+n(B\cap C)+n(C\cap A)=28$

$\Rightarrow (a+d)+(b+d)+(c+d)=28$

$\Rightarrow (a+b+c)+3d=28$

$\Rightarrow (a+b+c)+(3\times 7)=28$ $\{\because d=7\}$

$\Rightarrow (a+b+c)=7$

Hence, $7$ students receive medals in exactly $2$ sports.