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?
Solution
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. Therefore, the answer is 
which is equivalent to 28, 
Solution 2
We first figure out how many triangles can be created. This is done by choosing 
lines out of the 
, which is equivalent to 
. However, some of these triangles have vertices on the circle. Therefore, the answer choice must be less than 
. The only one that is so is .
See Also
| 2010 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 20 |
Followed by Problem 22 | |
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| All AMC 10 Problems and Solutions | ||
