Understanding the Handshake Problem I need help with this problem. The problem goes like this: In some countries it is customary to shake hands with everybody in the meeting. If there are two people there is 1 handshake, if there are three people there are three handshakes and so on. I know that the formula is $ \dfrac{n(n-1)}{2} $ but how do I get to this solution using a thinking process, specifically how would you solve this? Thanks.

Another take on the getting the same formula:
Rank the $n$ people in some defined way: age, salary, whatever.
Top person gets handshakes from $n-1$ people younger/poorer paid than him/her.
Next in the ordering gets $n-2$ handshakes from those "beneath" him/her, and so on.  Last person gets $0$ handshakes from underlings.
What is the sum of all the integers from $0$ to $n-1?$
Really just another way to avoid double counting...
