Question

In a sports fest, a school awarded 58 medals in athletics, 20 medals for indoor games and 25 medals for outdoor games. If these medals were bagged by a total of 78 students and only 5 students got medals for all the three activities, then the number of students who received medals for exactly two of the three activities is

• A. 30
• B. 15
• C. 20
• D. 40

Answer 1

The correct option is B 15

Let A, I, O be the set of students who bagged medals in athletics, indoor games and outdoor games respectively.
It is given that
$n\left(A\right)=58,n\left(I\right)=20,n\left(O\right)=25,n(A\cup I\cup O)=78\mathrm{and}n(A\cap I\cap O)=5.$
Let x denotes the number of students who got medals in athletics and indoor games only, y denotes the number of students who got medals in indoor games and outdoor games only, z denotes the number of students who got medals in athletics and outdoor games only.
This data can be represented in form of Venn Diagram as

Now,
$n(A\cup I\cup O)=78\Rightarrow n\left(A\right)+n\left(I\right)+n\left(O\right)-n(A\cap I)-n(I\cap O)-n(A\cap O)+n(A\cap I\cap O)=78\Rightarrow 58+20+25-n(A\cap I)-n(I\cap O)-n(A\cap O)+5=78\Rightarrow 108-n(A\cap I)-n(I\cap O)-n(A\cap O)=78\Rightarrow n(A\cap I)+n(I\cap O)+n(A\cap O)=30\Rightarrow \left(x+5\right)+\left(y+5\right)+\left(z+5\right)=30\Rightarrow x+y+z=15$
Therefore, required number of students is 15.