How many triangles are there with vertices from the points shown below? Note, we are not allowing degenerate triangles - ones with all three vertices on the same line, but we do allow non-right triangles. a. How many triangles are possible? triangles. Explain why your answer is correct. (Note: You will not receive credit until your instructor reviews your answer.) b. Find the number of triangles again, using a different method. Explain why your new method works. (Note: You will not receive credit until your instructor reviews your answer.) c. State a binomial identity that your two answers establish (that is, give the binomial identity that your two answers are a proof for.) Then generalize this using m's and n's. (Note: You will not receive credit until your instructor reviews your answer.) How many triangles can you draw using the dots below as vertices? triangles. a. Find an expression for the answer which is the sum of three terms involving binomial coefficients. (Note: You will not receive credit until your instructor reviews your answer.) b. Find an expression for the answer which is the difference of two binomial coefficients. (Note: You will not receive credit until your instructor reviews your answer.) c. Generalize the above to state and prove a binomial identity using a combinatorial proof. Say you have x points on the horizontal axis and y points in the semicircle. (Note: You will not receive credit until your instructor reviews your answer.)

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