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The Handshake Puzzle

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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

 

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

 

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?
hint

solution

 

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
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

 

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