4. How many triangles can you draw using the dots below as vertices? a. Find an expression for the answer which is the sum of three terms involving binomial coefficients. b. Find an expression for the answer which is the difference of two binomial coefficients. 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 semi-circle. Hint. There will be 185 triangles. But to find them... a. How many vertices of the triangle can be on the horizontal axis? b. Will any three dots work as the vertices?

4. How many triangles can you draw using the dots below as vertices? a. Find an expression for the answer which is the sum of three terms involving binomial coefficients. b. Find an expression for the answer which is the difference of two binomial coefficients. 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 semi-circle. Hint. There will be 185 triangles. But to find them... a. How many vertices of the triangle can be on the horizontal axis? b. Will any three dots work as the vertices?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 50E: Show that if the statement 1+2+3+...+n=n(n+1)2+2 is assumed to be true for n=k, the same equation...

See similar textbooks