Consider all the triangles you can create using the points shown below as vertices. Note, we...

 

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Consider all the triangles you can create using the points shown below as vertices. Note, we are not allowing degenerate triangles (ones with all three vertices on the same line) but we do allow non-right triangles. (a) Find the number of triangles, and explain why your answer is correct. (b) Find the number of triangles again, using a different method. Explain why your new method works. (c) State a binomial identity that your two answers above establish (that is, give the binomial identity that your two answers a proof for). Then generalize this using m's and n's. Consider all the triangles you can create using the points shown below as vertices. Note, we are not allowing degenerate triangles (ones with all three vertices on the same line) but we do allow non-right triangles. (a) Find the number of triangles, and explain why your answer is correct. (b) Find the number of triangles again, using a different method. Explain why your new method works. (c) State a binomial identity that your two answers above establish (that is, give the binomial identity that your two answers a proof for). Then generalize this using m's and n's.

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