In a group of children ,each child gives a gift to every other child ,If the number of gifts is 132 ,then the number of children are

Options

A. 11
B. 12
C. 14
D. None of these

Answer

To solve the problem of how many children are in a group where each child gives a gift to every other child, resulting in a total of 132 gifts, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
Each child gives a gift to every other child. If there are \( n \) children, each child gives gifts to \( n - 1 \) other children.

2. **Setting Up the Equation**:
The total number of gifts exchanged can be represented as:
\[ \text{Total Gifts} = n \times (n - 1) \]
Since each child gives a gift to \( n - 1 \) children, and there are \( n \) children.

3. **Equating to the Given Total**:
We know from the problem that the total number of gifts is 132. Therefore, we can set up the equation:
\[ n(n - 1) = 132 \]

4. **Rearranging the Equation**:
Rearranging gives us:
\[ n^2 - n - 132 = 0 \]

5. **Factoring the Quadratic Equation**:
We need to factor the quadratic equation \( n^2 - n - 132 = 0 \). We look for two numbers that multiply to \(-132\) and add to \(-1\). The numbers are \( -12 \) and \( 11 \):
\[ n^2 - 12n + 11n - 132 = 0 \]
This can be factored as:
\[ (n - 12)(n + 11) = 0 \]

6. **Finding the Values of \( n \)**:
Setting each factor to zero gives us:
\[ n - 12 = 0 \quad \text{or} \quad n + 11 = 0 \]
Thus, we find:
\[ n = 12 \quad \text{or} \quad n = -11 \]

7. **Determining the Valid Solution**:
Since the number of children cannot be negative, we discard \( n = -11 \). Therefore, the number of children is:
\[ n = 12 \]

### Final Answer:
The number of children is **12**.