Fifteen people meet, and each of them shakes hands once with every other person…

A combination is a selection in which order does not matter. $^{n}C_{r} = \dfrac{n!}{r!\,(n - r)!} = \dfrac{n(n - 1)\cdots(n - r + 1)}{r!}$, and $^{n}P_{r} = {}^{n}C_{r} \times r!$. Also $^{n}C_{0} = {}^{n}C_{n} = 1$ and $^{n}C_{2} = \dfrac{n(n - 1)}{2}$.