Handshakes

Question

Twenty people attend a meeting and everyone shakes hands with everyone else exactly once. How many handshakes occur?

Accepted Answer

Each handshake is a choice of 2 people from 20, order irrelevant: binom202 = frac20 cdot 192 = 190. Approach 2, double counting Each of 20 people shakes 19 hands → 20 times 19 = 380 person-shakes, but each handshake was counted twice (once per participant): 380 / 2 = 190. The divide-by-symmetry move in Approach 2 is the foundation for harder counting: whenever you count ordered things but want unordered ones, divide by the number of orderings.

Handshakes

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