Read the following information carefully to answer the questions that follow.
In a survey of 25 students, it was found that 15 have taken Mathematics. 12 have taken Physics and 11 taken Chemistry, 5 have taken Mathematics and Chemistry, 9 have taken Mathematics and Physics, 4 have taken Physics and Chemistry and 3 have taken all the three subjects.
The number of students who have taken only two subjects, is
(a)7
(b)8
(c)9
(d)10

Options

A. 7
B. 8
C. 9
D. 10

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion and Venn diagrams to find the number of students who have taken only two subjects.

### Step 1: Define the sets
Let:
- \( M \) = Number of students who have taken Mathematics = 15
- \( P \) = Number of students who have taken Physics = 12
- \( C \) = Number of students who have taken Chemistry = 11

### Step 2: Define the intersections
From the problem, we have:
- \( |M \cap C| \) = Number of students who have taken both Mathematics and Chemistry = 5
- \( |M \cap P| \) = Number of students who have taken both Mathematics and Physics = 9
- \( |P \cap C| \) = Number of students who have taken both Physics and Chemistry = 4
- \( |M \cap P \cap C| \) = Number of students who have taken all three subjects = 3

### Step 3: Calculate the number of students in each intersection
Now we will find the number of students who have taken only two subjects.

1. **Mathematics and Chemistry only**:
\[ |M \cap C| - |M \cap P \cap C| = 5 - 3 = 2 \]
(These are the students who took only Mathematics and Chemistry.)

2. **Mathematics and Physics only**:
\[ |M \cap P| - |M \cap P \cap C| = 9 - 3 = 6 \]
(These are the students who took only Mathematics and Physics.)

3. **Physics and Chemistry only**:
\[ |P \cap C| - |M \cap P \cap C| = 4 - 3 = 1 \]
(These are the students who took only Physics and Chemistry.)

### Step 4: Sum the students who have taken only two subjects
Now we will add the students who have taken only two subjects:
\[ \text{Total students who have taken only two subjects} = 2 + 6 + 1 = 9 \]

### Conclusion
The number of students who have taken only two subjects is **9**.

### Answer
(c) 9

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