Handshakes at the party

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At a party there are $15$ people and each shakes hands with every other exactly once. How many handshakes occur in all?

Solution

Each handshake corresponds to an unordered pair of people: the handshake between $X$ and $Y$ is the same as the one between $Y$ and $X$. The number of pairs is $N = \binom{15}{2} = \frac{15\cdot 14}{2} = \boxed{105}.$