There are n students in a class and a group of three is taken by class teacher to APPU ghar. The same set of three children is not taken more than once. It is found that she goes to Appu ghar 84 times more than the visit of a particular child , then the value of n is

Options

A. 10
B. 12
C. 60
D. none of these

Answer

To solve the problem, we need to find the number of students \( n \) in a class where a group of three students is taken to Appu Ghar, and it is given that the teacher goes to Appu Ghar 84 times more than the visits of a particular child.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The teacher takes groups of 3 students from a total of \( n \) students.
- Each group of 3 students is unique and not repeated.
- The total number of times the teacher goes to Appu Ghar is 84 times the number of times a particular child goes.

2. **Finding Total Groups of 3**:
- The number of ways to choose 3 students from \( n \) students is given by the combination formula:
\[ \binom{n}{3} = \frac{n!}{3!(n-3)!} \]

3. **Finding Groups Including a Particular Child**:
- If we fix one particular child (let's say the first child), we need to choose 2 more students from the remaining \( n-1 \) students.
- The number of ways to choose 2 students from \( n-1 \) is:
\[ \binom{n-1}{2} = \frac{(n-1)!}{2!(n-3)!} \]

4. **Setting Up the Equation**:
- The teacher goes to Appu Ghar a total of \( \binom{n}{3} \) times.
- The particular child goes to Appu Ghar \( \binom{n-1}{2} \) times.
- According to the problem, the teacher's visits are 84 times more than the child's visits:
\[ \binom{n}{3} = 84 \times \binom{n-1}{2} \]

5. **Substituting the Combinations**:
- Substitute the combinations into the equation:
\[ \frac{n(n-1)(n-2)}{6} = 84 \times \frac{(n-1)(n-2)}{2} \]

6. **Simplifying the Equation**:
- Cancel \( (n-1)(n-2) \) from both sides (assuming \( n > 2 \)):
\[ \frac{n}{6} = 42 \]
- Multiply both sides by 6:
\[ n = 252 \]

7. **Final Calculation**:
- Thus, the value of \( n \) is:
\[ n = 252 \]

### Conclusion:
The number of students in the class is \( n = 252 \).