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Person A shakes with nine people (B through J).Person B shakes with eight (C through J because he's already shaken hands with A). Person C shakes with seven, D with six, all the way down to person J, who has already shaken with everyone else. So, 9+8+7+6+5+4+3+2+1+0=45 shakes.

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Q: When ten people in a room shake hands with everyone else how many handshakes will there be in total?
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How many people would you need to guarantee that there were at least 1000 handshakes?

You could never guarantee 1000 handshakes because the people may choose not to shake hands! If each person did shake hands with everyone else, then 46 people would suffice.


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