# 2010 AMC 10A Problems/Problem 22

## Problem

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? $\textbf{(A)}\ 28 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 70 \qquad \textbf{(D)}\ 84 \qquad \textbf{(E)}\ 140$

## Solution 1

To choose a chord, we know that two points must be chosen. This implies that for three chords to create a triangle and not intersect at a single point, six points need to be chosen. We also know that for any six points we pick, there is only 1 way to connect the chords such that a triangle is formed in the circle's interior. Therefore, the answer is ${{8}\choose{6}}$, which is equivalent to $\boxed{\textbf{(A) }28}$.

## Solution 2 (Using the Answer Choices)

There are a total of $\dbinom{8}{3}=56$ total triangles that can be made out of these chords. We know that the amount of triangles which have all their vertices inside the circle has to be less than this, to the answer can only be $\boxed{\textbf{(A) }28}$. In general, the number of triangles that are in the interior will always be $\binom{n}{6}$ (can you prove it?)

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