In a class, 18 students took Physics, 23 students took Chemistry and 24 students took Mathematics of these 13 took both Chemistry and Mathematics, 12 took both Physics and Chemistry and 11 took both Physics and Mathematics. If 6 students offered all the three subjects, find:
(iii) How many took exactly one of the three subjects.

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion. We need to find out how many students took exactly one of the three subjects: Physics, Chemistry, and Mathematics.

### Step 1: Define the Variables
Let:
- \( n(P) = 18 \) (students who took Physics)
- \( n(C) = 23 \) (students who took Chemistry)
- \( n(M) = 24 \) (students who took Mathematics)
- \( n(C \cap M) = 13 \) (students who took both Chemistry and Mathematics)
- \( n(P \cap C) = 12 \) (students who took both Physics and Chemistry)
- \( n(P \cap M) = 11 \) (students who took both Physics and Mathematics)
- \( n(P \cap C \cap M) = 6 \) (students who took all three subjects)

### Step 2: Calculate the Number of Students in Each Intersection
Using the inclusion-exclusion principle, we can find the number of students in each intersection:

1. Students who took only Physics:
\[ n(P \text{ only}) = n(P) - (n(P \cap C) + n(P \cap M) - n(P \cap C \cap M)) \]
\[ n(P \text{ only}) = 18 - (12 + 11 - 6) = 18 - 17 = 1 \]

2. Students who took only Chemistry:
\[ n(C \text{ only}) = n(C) - (n(P \cap C) + n(C \cap M) - n(P \cap C \cap M)) \]
\[ n(C \text{ only}) = 23 - (12 + 13 - 6) = 23 - 19 = 4 \]

3. Students who took only Mathematics:
\[ n(M \text{ only}) = n(M) - (n(P \cap M) + n(C \cap M) - n(P \cap C \cap M)) \]
\[ n(M \text{ only}) = 24 - (11 + 13 - 6) = 24 - 18 = 6 \]

### Step 3: Calculate the Total Number of Students Who Took Exactly One Subject
Now, we can sum the number of students who took only one subject:
\[ n(\text{exactly one}) = n(P \text{ only}) + n(C \text{ only}) + n(M \text{ only}) \]
\[ n(\text{exactly one}) = 1 + 4 + 6 = 11 \]

### Final Answer
The number of students who took exactly one of the three subjects is **11**.

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