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
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:
\[
\mathbf{B. \ 65}
\]
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