Handshakes at a party

#handshake#edges#complete graph

Statement

If $10$ people at a party all shake hands with each other, how many handshakes occur?

Hint

Each pair contributes one handshake: $\binom{10}{2}$ .

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Count pairs. $\binom{10}{2}=\frac{10\cdot9}{2}=45$ .

Conclusion. There are $45$ handshakes.