Spring 2016 COMP 2300 Discrete Structures for Computation Chapter 9.7 Pascal’s Formula and the Binomial Theorem Donghyun (David) Kim Department of Mathematics and Computer Science North Carolina Central University

Pascal’s Formula Pascal’s formula: Let n and r be positive integers and suppose . Then, why? Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

Pascal’s Formula – cont’ Pascal’s formula: Let n and r be positive integers and suppose . Then, Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

Pascal’sTriangle Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

Deriving New Formulas from Pascal’s Formula Use Pascal’s formula to derive a formula for in terms of values of , , and . Assume n and r are nonnegative integers and . Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

Deriving New Formulas from Pascal’s Formula – cont’ Use Pascal’s formula to derive a formula for in terms of values of , , and . Assume n and r are nonnegative integers and . Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

The Binomial Theorem In algebra a sum of two terms, such as , is called a binomial. The binomial theorem gives an expression for the powers of a binomial , for each positive integer n and all real numbers a and b. Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

The Binomial Theorem – cont’ Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

The Binomial Theorem – cont’ Proof of the Binomial Theorem (By Math. Induction) When n=1, Suppose Then, we have Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

Deriving Another Combinatorial Identity from the Binomial Theorem Use the binomial theorem to show that for all integers Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

Deriving Another Combinatorial Identity from the Binomial Theorem Use the binomial theorem to show that for all integers Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

Using a Combinatorial Argument to Derive the Identity Show that Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

Using a Combinatorial Argument to Derive the Identity – cont’ Show that Suppose S is a set with n elements. Number of subsets of S Number of subsets of size 0 Number of subsets of size 1 Number of subsets of size n Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

Using a Combinatorial Argument to Derive the Identity – cont’ Show that Suppose S is a set with n elements. Number of subsets of S Number of subsets of size 0 Number of subsets of size 1 Number of subsets of size n Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

Using the Binomial Theorem to Simplify a Sum Express the following sum in closed form (without using a summation symbol and without using an ellipsis…): Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

Using the Binomial Theorem to Simplify a Sum – cont’ Express the following sum in closed form (without using a summation symbol and without using an ellipsis…): Spring 2016 COMP 2300 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University