Handshake Puzzle

This brain bender is going to go in stages -- each getting harder than the last... See how far you can go!
Stage 1:
If you have TWO people in a room and each person shakes hands with every other person exactly once, how many total handshakes happen? Solution > Handshake Puzzle Stage 1
Stage 2:
If you have THREE people in a room and each person shakes hands with every other person exactly once, how many total handshakes happen? Solution > Handshake Puzzle Stage 2
Stage 3:
If you have FOUR people in a room and each person shakes hands with every other person exactly once, how many total handshakes happen?Solution > Handshake Puzzle Stage 3
Stage 4:
If you have FIVE people in a room and each person shakes hands with every other person exactly once, how many total handshakes happen? Solution > Handshake Puzzle Stage 4
Stage 5:
So, if you have SIX people in a room and each person shakes hands with every other person exactly once, how many total handshakes happen? SEVEN people?
EIGHT people?
Solution > Handshake Puzzle Stage 5