Answer
The number of students that like only two of the activities are 34
Step-by-step explanation:
Number of students that enjoy video games, A = 38
Number of students that enjoy going to the movies, B = 12
Number of students that enjoy solving mathematical problems, C = 24
A∩B∩C = 8
Here we have;
n (A∪B∪C) = n (A) + n (B) + n (C) - n (A∩B) - n (B∩C) - n (A∩C) + n (A∩B∩C)
= 38 + 12 + 24 - n (A∩B) - n (B∩C) - n (A∩C) + 8
Also the number of student that like only one activity is found from the following equation;
n (A) - n (A∩B) - n (A∩C) + n (A∩B∩C) + n (B) - n (A∩B) - n (B∩C) + n (A∩B∩C) + n (C) - n (C∩B) - n (A∩C) + n (A∩B∩C) = 30
n (A) + n (B) + n (C) - 2·n (A∩B) - 2·n (A∩C) - 2·n (B∩C) + 3·n (A∩B∩C) = 30
38 + 12 + 24 - 2·n (A∩B) - 2·n (A∩C) - 2·n (B∩C) + 24 = 30
- 2·n (A∩B) - 2·n (A∩C) - 2·n (B∩C) = - 68
n (A∩B) + n (B∩C) + n (A∩C) = 34
Therefore, the number of students that like only two of the activities = 34.
