At a party, everyone shook hands with everybody else. There were 66 handshakes. How many people were at the party?

At a party, everyone shook hands with everybody else. There were 66 handshakes. How many people were at the party?
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Answer :

With two people, there is one handshake. With three people, there are three handshakes. With four people, there are six handshakes. In general, with n+1 people, the number of handshakes is the sum of the first n consecutive numbers: 1+2+3+...+n. Since this sum is n(n+1)/2, we need to solve the equation n(n+1)/2 = 66. This is the quadratic equation n2 +n-132 = 0. Solving for n, we obtain 11 as the answer and deduce that there were 12 people at the party.
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