2005 AMC 8 Problems/Problem 21

Problem

How many distinct triangles can be drawn using three of the dots below as vertices?

$[asy]dot(origin^^(1,0)^^(2,0)^^(0,1)^^(1,1)^^(2,1));[/asy]$

$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$

Solution

The number of ways to choose three points to make a triangle is $\binom 63 = 20$ . However, two* of these are a straight line so we subtract $2$ to get $\boxed{\textbf{(C)}\ 18}$ .

Note: We are assuming that there are no degenerate triangles in this problem, and that is why we subtract two.

Video solution

https://www.youtube.com/watch?v=XQS-KVW1O6M ~David

2005 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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2005 AMC 8 Problems/Problem 21

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Problem

Solution

Video solution

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