Every two persons shakes hands with each other in a party and the total number of handis 66. The number of guests in the party is

Options

A. 11
B. 12
C. 13
D. 14

Answer

To solve the problem of finding the number of guests in a party where every two persons shake hands and the total number of handshakes is 66, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We know that every two persons shake hands. If there are \( n \) guests, the total number of handshakes can be calculated using the combination formula \( C(n, 2) \), which is given by:
\[ C(n, 2) = \frac{n(n-1)}{2} \]
- According to the problem, the total number of handshakes is 66.

2. **Setting Up the Equation**:
- We can set up the equation based on the information given:
\[ \frac{n(n-1)}{2} = 66 \]

3. **Eliminating the Fraction**:
- To eliminate the fraction, multiply both sides of the equation by 2:
\[ n(n-1) = 132 \]

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

5. **Factoring the Quadratic**:
- We can factor the quadratic equation:
\[ n^2 - 12n + 11n - 132 = 0 \]
- This can be factored as:
\[ (n - 12)(n + 11) = 0 \]

6. **Finding the Roots**:
- 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. **Interpreting the Results**:
- Since the number of guests cannot be negative, we discard \( n = -11 \).
- Therefore, the number of guests in the party is:
\[ n = 12 \]

### Final Answer:
The number of guests in the party is **12**.