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

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.

2010 AMC 10A Problem 22

Answer: A

Difficulty rating: 2160

Solution:

Problem 22 in Other Years