In the final examinations for `9^(th)` grade students of New York Public School, 4 students failed in Mathematics Sceince and History, 16 students failed in at least two of the above - mentioned subjects and 25 students failed in at least one of the above subject . How many students failed in exactly one of the mentioned subjects ?

Options

A. 5
B. 7
C. 9
D. 13

Answer

To solve the problem step by step, we will define variables and set up equations based on the information provided in the question.

### Step 1: Define Variables
Let:
- \( f_1 \) = number of students who failed in exactly 1 subject
- \( f_2 \) = number of students who failed in exactly 2 subjects
- \( f_3 \) = number of students who failed in exactly 3 subjects

### Step 2: Analyze the Information Given
From the problem statement:
1. **4 students failed in Mathematics, Science, and History**: This means \( f_3 = 4 \).
2. **16 students failed in at least 2 subjects**: This includes those who failed in exactly 2 subjects and those who failed in all 3 subjects. Therefore, we can write:
\[ f_2 + f_3 = 16 \]
3. **25 students failed in at least 1 subject**: This includes those who failed in exactly 1 subject, exactly 2 subjects, and exactly 3 subjects. Thus, we can write:
\[ f_1 + f_2 + f_3 = 25 \]

### Step 3: Set Up the Equations
From the information, we have the following equations:
1. \( f_3 = 4 \) (from the first statement)
2. \( f_2 + 4 = 16 \) (from the second statement)
3. \( f_1 + f_2 + 4 = 25 \) (from the third statement)

### Step 4: Solve for \( f_2 \)
Using the second equation:
\[ f_2 + 4 = 16 \]
Subtract 4 from both sides:
\[ f_2 = 16 - 4 = 12 \]

### Step 5: Solve for \( f_1 \)
Now, substitute \( f_2 \) into the third equation:
\[ f_1 + 12 + 4 = 25 \]
Combine the constants:
\[ f_1 + 16 = 25 \]
Subtract 16 from both sides:
\[ f_1 = 25 - 16 = 9 \]

### Conclusion
The number of students who failed in exactly 1 subject is \( f_1 = 9 \).

### Final Answer
Thus, the answer is **9 students** failed in exactly one of the mentioned subjects.

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