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

Step-by-step explanation:

Other answers have already shown that there are n=12 people at the party, either using combinatorial choose notation

(n2) = n!2! (n-2) !=66

or by counting handshakes person by person, noting that each successive person has one fewer unique handshake than the prior (as the previously counted people are disregarded), and then using the sum-of-consecutive-integers formula to get

(n-1) + (n-2) + ... + 2+1=∑n-1k=1k = (n-1) n2=66

and solving for n.

Here I will provide another method of arriving at n (n-1) 2=66.

Instead of keeping track of unique handshakes from the start, note that each person shakes hands with n-1 other people. As there are n people at the party, this gives us n (n-1) handshakes. However, notice that we counted each handshake twice; for example, if Alice and Bob shook hands, we counted that once for Alice, and then again for Bob. To account for the double-counting, we can divide by 2, which gives us the previously mentioned formula of n (n-1) 2.

As a side note, if we did not previously know the sum formula, we could use this as a combinatorial proof to show that it works. Also, if we account for repeated counting by dividing by k!, the number of permutations of k objects, a similar method can be used to derive the chosen formula (nk) = n! k! (n-k) !.

There were 12 people at the party