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 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?

Explanation

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
g + f = 20 implies f = 10 `[because g = 10]`
g + e = 30 implies e = 20
d + g = 25 implies d = 15
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