1989 AIME Problem 2

Question

Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices?

Answer

Answer: 968

Concepts:combinations complementary counting subsets

Difficulty rating: 1780

Small hint:

Each choice of at least three marked points determines one convex polygon

Big hint:

Count all subsets and remove those of sizes

0,

1,

and

2

Solution:

Every subset of at least three points determines exactly one convex polygon. There are

2^{10}=1024

subsets in all. Of these,

\binom{10}{0}=1,

\binom{10}{1}=10,

and

\binom{10}{2}=45

have fewer than three points. Therefore the required number is

1024-1-10-45=968.