After a business meeting, all attendees shook hands with every other person at the meeting. If there were a total of 66 handshakes, how many people were at the meeting?

To solve this problem, we need to find the number of people at a meeting where each person shakes hands with every other person exactly once, resulting in a total of 66 handshakes.

### Step-by-Step Solution

1. **Understand the Problem**:  
   If there are \( n \) people at the meeting, each person shakes hands with every other person exactly once. The number of ways to choose 2 people out of \( n \) to shake hands is given by the combination formula:
   \[
   \binom{n}{2} = \frac{n(n-1)}{2}
   \]
   This expression represents the total number of unique handshakes.

2. **Set Up the Equation**:  
   We know the total number of handshakes is 66. Therefore, we set up the equation:
   \[
   \frac{n(n-1)}{2} = 66
   \]

3. **Solve for \( n \)**:  
   Multiply both sides by 2 to clear the fraction:
   \[
   n(n-1) = 132
   \]

   Rewrite the equation:
   \[
   n^2 - n - 132 = 0
   \]

4. **Factor 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 up to \(-1\). These numbers are 12 and -11.

   The equation factors as:
   \[
   (n - 12)(n + 11) = 0
   \]

5. **Find the Valid Solution for \( n \)**:  
   The solutions to the equation are:
   \[
   n - 12 = 0 \quad \text{or} \quad n + 11 = 0
   \]

   Solving these gives:
   \[
   n = 12 \quad \text{or} \quad n = -11
   \]

   Since \( n \) represents the number of people, it must be a positive integer. Thus, \( n = 12 \).

### Conclusion

There were **12 people** at the meeting.

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Would you like more details, or do you have any questions?

Here are 5 related questions to consider:

1. What is the general formula for the number of handshakes if there are \( n \) people at a meeting?
2. How many handshakes occur in a meeting with 15 people?
3. If each person shakes hands with every other person twice, how would the formula change?
4. What if a certain number of handshakes did not occur? How would we adjust the calculation?
5. How can we derive a formula to calculate the number of people given a specific number of handshakes?

**Tip:** Remember that the combination formula \(\binom{n}{k}\) can be used to calculate the number of ways to choose \( k \) items from \( n \) without regard to the order of selection.

Discover how to calculate the number of people at a meeting where each person shakes hands exactly once with every other person, resulting in 66 handshakes. Step-by-step solution provided using combination and quadratic equations. Suitable for Grades 7-9.

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