IN a meeting everyone had shaken hands with everyone else, it was found that 66 handshakes were exchanged . Number of persons present in the meeting is

Options

A. 17
B. 12
C. 13
D. 18

Answer

To solve the problem of finding the number of persons present in a meeting where 66 handshakes were exchanged, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that in a meeting, every person shakes hands with every other person. The total number of handshakes is given as 66.

2. **Setting Up the Equation**:
Let the total number of persons in the meeting be denoted by \( n \). The number of ways to choose 2 persons from \( n \) persons (which represents one handshake) is given by the combination formula \( \binom{n}{2} \). This can be expressed as:
\[ \binom{n}{2} = \frac{n(n-1)}{2} \]
According to the problem, this is equal to 66:
\[ \frac{n(n-1)}{2} = 66 \]

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 are -12 and 11. Thus, we can factor the equation as:
\[ (n - 12)(n + 11) = 0 \]

6. **Finding the Roots**:
Setting each factor to zero gives us:
\[ n - 12 = 0 \quad \text{or} \quad n + 11 = 0 \]
This results in:
\[ n = 12 \quad \text{or} \quad n = -11 \]

7. **Interpreting the Results**:
Since \( n \) represents the number of persons, it cannot be negative. Therefore, we discard \( n = -11 \).

8. **Final Answer**:
The number of persons present in the meeting is:
\[ n = 12 \]