In a survey where 100 students reported which subjects they like, 32 students in total liked Mathematics, 38 students liked Business and 30 students liked Literature. Moreover 7 students liked both Mathematics and Literature, 10 students liked both Mathematics and Business, 8 students liked both Business and Literature, 5 students liked all three subjects. Then the number of people who liked exactly one subject is
Step-by-step Solution:
To find the number of students who liked exactly one subject, we use the principle of set operations.
We define:
- \( M \) as the set of students who like Mathematics.
- \( B \) as the set of students who like Business.
- \( L \) as the set of students who like Literature.
Given data:,br>
- \( |M| = 32 \)
- \( |B| = 38 \)
- \( |L| = 30 \)
- \( |M \cap L| = 7 \) (students who like both Mathematics and Literature)
- \( |M \cap B| = 10 \) (students who like both Mathematics and Business)
- \( |B \cap L| = 8 \) (students who like both Business and Literature)
- \( |M \cap B \cap L| = 5 \) (students who like all three subjects)
We calculate the number of students who like exactly one subject:
1. Students who like only Mathematics:
\[
|M| - |M \cap L| - |M \cap B| + |M \cap B \cap L|
\]
\[
= 32 - 7 - 10 + 5 = 20
\]
2. Students who like only Business:
\[
|B| - |M \cap B| - |B \cap L| + |M \cap B \cap L|
\]
\[
= 38 - 10 - 8 + 5 = 25
\]
3. Students who like only Literature:
\[
|L| - |M \cap L| - |B \cap L| + |M \cap B \cap L|
\]
\[
= 30 - 7 - 8 + 5 = 20
\]
Total students who liked exactly one subject:
\[
20 + 25 + 20 = 65
\]
Answer:
\[
\mathbf{B. \ 65}
\]