AIME 1989 · 第 2 题

Solution

Any subset of the ten points with three or more members can be made into exactly one such polygon. Thus, we need to count the number of such subsets. There are $2^{10} = 1024$ total subsets of a ten-member set, but of these ${10 \choose 0} = 1$ have 0 members, ${10 \choose 1} = 10$ have 1 member and ${10 \choose 2} = 45$ have 2 members. Thus the answer is $1024 - 1 - 10 - 45 = \boxed{968}$.

Note ${n \choose 0}+{n \choose 1} + {n \choose 2} + \dots + {n \choose n}$ is equivalent to $2^n$