1. 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

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

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

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

5. 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 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

6. 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 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

7. 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 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

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

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

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

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

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

13. 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 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

14. 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 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

15. 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 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University

16. 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 Department of Mathematics and Computer Science Donghyun (David) Kim North Carolina Central University