A college auiarded 38 medals in Football, 15 medals 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. Tile number of students who received medals in exactly two of the three sports is

Let F, B and C denote the sets of students who received medals in Football, Basketball and Cricket respectively. Then, we have 

$\begin{array}{l}n\left(F\right)=37,n\left(B\right)=15,n\left(C\right)=20\\ n(F\cup B\cup C)=58\text{ and }n(F\cap B\cap C)=3\end{array}$

Now,

$\begin{array}{l}n(F\cup B\cup C)=n\left(F\right)+n\left(B\right)+n\left(C\right)-n(F\cap B)-n(B\cap C)\\ -n(C\cap F)+n(F\cap B\cap C)\\ \Rightarrow 58=38+15+20-\left\{n\right(F\cap B)+n(B\cap C)+n(C\cap F)\mid +3\\ \Rightarrow n(F\cap B)+n(B\cap C)+n(C\cap F)=18\end{array}$

Hence, the number of students who received medals in exactly two of the three sports

$\begin{array}{l}=n(F\cap B)+n(B\cap C)+n(C\cap F)-3n(F\cap B\cap C)\\ =18-3\times 3=9.\end{array}$