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.