Answer:
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)!.