Exercise 3.

How many triangles can you draw using the dots below as vertices?

Twelve dots arranged in a half circle. Seven dots lie on the horizontal diameter of the circle, the remaining five lie on the circumference of the circle.

  1. Find an expression for the answer which is the sum of three terms involving binomial coefficients.

  2. Find an expression for the answer which is the difference of two binomial coefficients.

  3. 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 semi-circle.

Hint.
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