2010 AMC 10A Problem 22

Question

Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?

28

56

70

84

140

Answer: A

Concepts:counting intersections combinations bijection

Difficulty rating: 2160

Small hint:

Choose the six circle points used as endpoints of the three chords

Big hint:

For six points in order, only the three opposite-pair chords make the interior triangle

Solution:

An interior triangle is formed by three chords that pairwise intersect inside the circle. Such a triangle uses six distinct endpoints on the circle.

Conversely, for any six chosen points in circular order, exactly one set of three chords pairs opposite endpoints so that the three chords intersect pairwise inside the circle.

Therefore the number of triangles is

\binom{8}{6}=\binom{8}{2}=28.

Thus, A is the correct answer.