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.
Consider the following statements
I. The number of students who have taken only one subject is equal to the number of students who have taken only two subjects.
II. The number of students who have taken at least two subjects is four times the number of students who have taken all the three subjects.
Which of the above statement(s) is/are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To solve the problem, we will analyze the given data step by step and verify the two statements.

### Given Data:
- Total students surveyed = 25
- Students taking Mathematics (M) = 15
- Students taking Physics (P) = 12
- Students taking Chemistry (C) = 11
- Students taking both Mathematics and Chemistry (M ∩ C) = 5
- Students taking both Mathematics and Physics (M ∩ P) = 9
- Students taking both Physics and Chemistry (P ∩ C) = 4
- Students taking all three subjects (M ∩ P ∩ C) = 3

### Step 1: Calculate Students Taking Only One Subject
1. **Calculate students taking only Mathematics (M only)**:
\[ M \text{ only} = M - (M \cap P) - (M \cap C) + (M \cap P \cap C) \]
\[ M \text{ only} = 15 - 9 - 5 + 3 = 4 \]

2. **Calculate students taking only Physics (P only)**:
\[ P \text{ only} = P - (P \cap M) - (P \cap C) + (M \cap P \cap C) \]
\[ P \text{ only} = 12 - 9 - 4 + 3 = 2 \]

3. **Calculate students taking only Chemistry (C only)**:
\[ C \text{ only} = C - (C \cap M) - (C \cap P) + (M \cap P \cap C) \]
\[ C \text{ only} = 11 - 5 - 4 + 3 = 5 \]

### Step 2: Total Students Taking Only One Subject
\[ \text{Total (only one subject)} = M \text{ only} + P \text{ only} + C \text{ only} = 4 + 2 + 5 = 11 \]

### Step 3: Calculate Students Taking Only Two Subjects
1. **Calculate students taking Mathematics and Physics only (M ∩ P only)**:
\[ M \cap P \text{ only} = (M \cap P) - (M \cap P \cap C) = 9 - 3 = 6 \]

2. **Calculate students taking Mathematics and Chemistry only (M ∩ C only)**:
\[ M \cap C \text{ only} = (M \cap C) - (M \cap P \cap C) = 5 - 3 = 2 \]

3. **Calculate students taking Physics and Chemistry only (P ∩ C only)**:
\[ P \cap C \text{ only} = (P \cap C) - (M \cap P \cap C) = 4 - 3 = 1 \]

### Step 4: Total Students Taking Only Two Subjects
\[ \text{Total (only two subjects)} = M \cap P \text{ only} + M \cap C \text{ only} + P \cap C \text{ only} = 6 + 2 + 1 = 9 \]

### Step 5: Verify the Statements
1. **Statement I**: The number of students who have taken only one subject is equal to the number of students who have taken only two subjects.
- Only one subject = 11
- Only two subjects = 9
- **Result**: False (11 ≠ 9)

2. **Statement II**: The number of students who have taken at least two subjects is four times the number of students who have taken all three subjects.
- Students taking at least two subjects = (M ∩ P only) + (M ∩ C only) + (P ∩ C only) + (M ∩ P ∩ C) = 6 + 2 + 1 + 3 = 12
- Four times the number of students taking all three subjects = 4 × 3 = 12
- **Result**: True (12 = 12)

### Conclusion
- **Statement I** is false.
- **Statement II** is true.

### Answer
The correct option is **B**: Only the second statement is correct.