An investigator interviewed 100 students to determine their preferences for the three drinks; milk (M), coffee(C) and tea(T). He reported the following : 10 students has all the three drinks M, C, T; 20 had M and C; 30 had C and T, 25 had M and T; 12 had M only; 5 had C only and 8 had T only. Using a Venn diagram, find how many did not take any of the three drinks?

Given,  $M,C$  and   $T$  are the sets of drinks; milk, coffee and tea, respectively. Let us denote the number of drinks (students) contained in the bounded region as shown in the diagram by  $a,b,c,d,e,f$  and   $g$, respectively.
 
Then,     $g=10$
   $$\begin{gathered} g+f=20\Rightarrow f=10 [\because g=10] \\ g+e=30\Rightarrow e=20 \\ d+g=25\Rightarrow d=15 \end{gathered}$$
and         $a=12,b=5,c=8$
Thus, total number of students taking drinks  $M$  or  $C$   or  $T$
          $=a+b+c+d+e+f+g$
         $=12+5+8+15+20+10+10=80$
Hence, the number of students taking none of them drinks
         $=100-80=20$