Suppose there are 20 people in a room. each of those 20 shakes hands exactly once with everyone else. how many handshakes take place?

Only 20, because it takes 2 hands to shake.

Picture a smaller scenario to set up the problem correctly. let's say 4 people, A,B,C,D. The first person A shakes hands with the other 3. The second person B shakes hands with the other 2 since they've already shook hands with A. Third person C shakes hands with fourth person D. So the hand shake combos are (AB,AC,AD,BC,BD,CD) = 3+2+1 = 6. This is a combination, order does not matter. AC is the same as CA. So the answer is 19+18+17+16+15+14+13...+1 = 20nCr2 = 190

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