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)!.

we need more info to solve this

is there answer choices?