Everybody in a room shake hands with everybody else. If the total number of handshakes is 66, then the number of persons in the room is-

Options

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

Answer

To solve the problem of finding the number of persons in a room where everyone shakes hands with everyone else, we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Problem**:
We know that each handshake involves two people. If there are \( n \) people in the room, the total number of unique handshakes can be calculated using the combination formula \( C(n, 2) \), which represents the number of ways to choose 2 people from \( n \).

2. **Setting Up the Equation**:
The formula for combinations is given by:
\[ C(n, 2) = \frac{n(n-1)}{2} \]
According to the problem, the total number of handshakes is 66. Therefore, we can set up the equation:
\[ \frac{n(n-1)}{2} = 66 \]

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

4. **Rearranging the Equation**:
Rearranging the equation 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 -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 find:
\[ n = 12 \quad \text{or} \quad n = -11 \]

7. **Interpreting the Results**:
Since the number of persons cannot be negative, we discard \( n = -11 \). Therefore, the only valid solution is:
\[ n = 12 \]

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

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