user
At a party, every guest shook hands with every other guest exactly once. If there were 66 handshakes in total, how many guests were at the party? Explain your reasoning.
assistant
This is a classic combinatorics problem. Here's how to solve it:

1. Every pair of guests results in one handshake. So, the number of handshakes is the number of ways to pick 2 guests from the total number of guests.
2. This is a combination problem, represented mathematically as C(n, 2), where n is the total number of guests.
3. The formula for combinations is C(n, k) = n! / [k!(n-k)!]. In this case, we're looking for C(n, 2) = n*(n-1)/2.
4. We know that this equals 66, so setting up the equation gives us n*(n-1)/2 = 66.
5. Solving this equation for n requires some trial and error or knowledge of quadratic equations. The solution is n=12.

So, there were 12 guests at the party.