I am struggling with the following problem.
Suppose there are two distinct parties having the same number of people. In the first party, each person shakes hands with every other person in that party exactly once. Then one person leaves and everyone remaining shakes hands with everyone else once again exactly once. This procedure continues (i.e. one person leaves after which the remaining people shake hands again) until there are two people remaining, who shake hands once.
In the second party, a different handshake game is played. One person is selected to be VIP and every non-VIP party member shakes hands with the VIP exactly once. The VIP then leaves the party with one other person and two new people enter the party and both become VIPs. All non-VIPs shake hands with each of these two VIPs exactly once (the VIPs do not shake with each other). Then the two VIPs leave with one other person and three new people enter the party and become VIPs, and every non-VIP at the party shakes with each VIP exactly once (VIPs do not shake with each other). This process continues until only one person remains and shakes with each of the new VIPs (who are one less in number than the original people at the party).
If 165 total handshakes occurred in the the second party, how many total handshakes occurred in the first party?
I am lost in how to approach this problem, particularly in expressing the number of handshakes in the first party. However, for the second party, I gather that the total number of handshakes given $N$ people initially present, expressed as a dot product between two vectors, is $(1, 2,...,N-1) \cdot (N-1, N-2,...,1)$, which must equal 165. I am not sure how to solve for $N$ and approach the first party.