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. Then the number of students who received medals in exactly two of the three sports, is

• A. $18$
• B. $15$
• C. $9$
• D. $6$

Answer 1

The correct option is C $9$

Let $F,B,C$ be the sets of students who received medals in football, basketball, and cricket respectively.

$n\left(F\right)=38n\left(B\right)=15n\left(C\right)=20$

$n(F\cup B\cup C)=58n(F\cap B\cap C)=3$

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

$n(F\cap B)+n(B\cap C)+n(C\cap F)=73-58+3=18$

the number of students who received medals in exactly two of the three

sports is,

$n(F\cap B)+n(B\cap C)+n(C\cap F)-3\times n(F\cap B\cap C)=18-9=9$