Everybody in a room shakes hands with everybody else. The total number of hand shakes is 66. the total number of persons in the room is

Options

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

Answer

To solve the problem of finding the total number of persons in a room where everybody shakes hands with everybody else, and the total number of handshakes is given as 66, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to determine the number of persons (let's denote it as \( N \)) in a room where each person shakes hands with every other person. The total number of handshakes is given as 66.

2. **Using Combinations**:
The number of ways to choose 2 persons from \( N \) persons to shake hands is given by the combination formula \( \binom{N}{2} \). This can be expressed mathematically as:
\[ \binom{N}{2} = \frac{N(N-1)}{2} \]
Since the total number of handshakes is 66, we can set up the equation:
\[ \frac{N(N-1)}{2} = 66 \]

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 need to factor the quadratic equation. We look for two numbers that multiply to -132 and add to -1. The numbers -12 and 11 work:
\[ (N - 12)(N + 11) = 0 \]

6. **Finding the Roots**:
Setting each factor to zero gives us:
\[ N - 12 = 0 \quad \text{or} \quad N + 11 = 0 \]
Thus, we have:
\[ N = 12 \quad \text{or} \quad N = -11 \]

7. **Selecting the Valid Solution**:
Since the number of persons cannot be negative, we discard \( N = -11 \). Therefore, the total number of persons in the room is:
\[ N = 12 \]

### Final Answer:
The total number of persons in the room is **12**.

---