In a company, each employee gives a gift to every other employee. If the number of gifts is 56, then the number of employees in the company is :

Options

A. 11
B. 13
C. 12
D. 8

Answer

To solve the problem, we need to determine the number of employees in a company based on the number of gifts exchanged. Each employee gives a gift to every other employee, which means that if there are \( n \) employees, each employee gives \( n-1 \) gifts (to every other employee).

1. **Understanding the Gift Exchange**:
Each employee gives a gift to every other employee. Therefore, the total number of gifts given can be calculated using the formula for combinations, specifically \( C(n, 2) \), which represents the number of ways to choose 2 employees from \( n \) to exchange gifts. This is given by the formula:
\[ C(n, 2) = \frac{n(n-1)}{2} \]

2. **Setting Up the Equation**:
According to the problem, the total number of gifts exchanged is 56. Therefore, we can set up the equation:
\[ \frac{n(n-1)}{2} = 56 \]

3. **Solving for \( n \)**:
To eliminate the fraction, we can multiply both sides of the equation by 2:
\[ n(n-1) = 112 \]

4. **Rearranging the Equation**:
Rearranging gives us a quadratic equation:
\[ n^2 - n - 112 = 0 \]

5. **Applying the Quadratic Formula**:
We can solve this quadratic equation using the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -1 \), and \( c = -112 \). Plugging in these values:
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-112)}}{2 \cdot 1} \]
\[ n = \frac{1 \pm \sqrt{1 + 448}}{2} \]
\[ n = \frac{1 \pm \sqrt{449}}{2} \]

6. **Calculating the Square Root**:
The square root of 449 is approximately 21.2. Thus:
\[ n = \frac{1 \pm 21.2}{2} \]
This gives us two potential solutions:
\[ n = \frac{22.2}{2} \approx 11.1 \quad \text{(not valid since n must be an integer)} \]
\[ n = \frac{-20.2}{2} \quad \text{(not valid since n must be positive)} \]

7. **Finding Integer Solutions**:
Since \( n \) must be a whole number, we can try integer values for \( n \) to find the correct one:
- For \( n = 12 \):
\[ \frac{12 \cdot 11}{2} = 66 \quad \text{(too high)} \]
- For \( n = 11 \):
\[ \frac{11 \cdot 10}{2} = 55 \quad \text{(too low)} \]
- For \( n = 10 \):
\[ \frac{10 \cdot 9}{2} = 45 \quad \text{(too low)} \]
- For \( n = 8 \):
\[ \frac{8 \cdot 7}{2} = 28 \quad \text{(too low)} \]
- For \( n = 9 \):
\[ \frac{9 \cdot 8}{2} = 36 \quad \text{(too low)} \]
- For \( n = 7 \):
\[ \frac{7 \cdot 6}{2} = 21 \quad \text{(too low)} \]
- For \( n = 8 \):
\[ \frac{8 \cdot 7}{2} = 28 \quad \text{(too low)} \]
- For \( n = 12 \):
\[ \frac{12 \cdot 11}{2} = 66 \quad \text{(too high)} \]
- For \( n = 11 \):
\[ \frac{11 \cdot 10}{2} = 55 \quad \text{(too low)} \]
- For \( n = 10 \):
\[ \frac{10 \cdot 9}{2} = 45 \quad \text{(too low)} \]

After checking values, we find that \( n = 8 \) gives us a total of 28 gifts, which is not correct.

**Final Check**:
The correct integer solution for \( n \) that satisfies the equation \( n(n-1) = 112 \) is \( n = 16 \).

Thus, the number of employees in the company is **16**.