Question

A school awarded $30$ medals in tennis, $14$ in carrom and $25$ in badminton. If these medals were bagged by a total of $50$ students and only $5$ students got medals in all the three sports, then the number of medals received by students in exactly two of the three games is

• A. $30$
• B. $24$
• C. $9$
• D. $18$

Answer 1

The correct option is C $9$

Let $T,C,B$ denote the sets of students who bagged medals in tennis, carrom and badminton respectively.
$\Rightarrow n\left(T\right)=30;n\left(C\right)=14;n\left(B\right)=25$
Also, $n(T\cup C\cup B)=50;n(T\cap C\cap B)=5$
Now, we know the formula:
$n(T\cup C\cup B)=n\left(T\right)+n\left(C\right)+n\left(B\right)$
$-n(T\cap C)-n(C\cap B)-n(B\cap T)$
$+n(C\cap B\cap T)$
To find: Number of medals received by students in exactly $2$ games.
$n(T\cap C)+n(T\cap B)+n(B\cap C)=x\left(let\right)$
Inserting the values in the formula:
$50=30+14+25-x+5$
$\Rightarrow x=30+14+25+5-50=24$
Number of medals received by students in exactly $2$ games=$n(T\cap C)+n(T\cap B)+n(B\cap C)-3n(T\cap B\cap C)=24-3\left(5\right)=9$
Hence, $9$ students bagged exactly two out of three medals.