1. MAT 4830 Numerical Analysis Binomial Coefficients and Combinatorial Identities http://myhome.spu.edu/lauw

2. Goals Binomial Theorem Binomial Coefficients Combinatorial Identities Review shifting indices Review Induction

3. Take Home Exam Need Binomial Coefficients for the second problem. Need Binomial Theorem for a few parts of the second problem.

4. Binomial Expansion

5. Binomial Theorem

6. Useful Formulas for Binomial Coefficients

7. Pascal’s Identity

8. Proof:Analysis

9. Binomial Theorem Combinatorial Proof:Analysis

10. Example 1

11. Example 2 Proof:Analysis

12. Example 3 (a) Proof: 1. Induction 2. Can be done without induction, but need to take care special cases. Analysis

13. Example 3 (b) Solution:Analysis

14. Binomial Theorem Induction Proof: Need some preparations Analysis

15. Binomial Theorem Proof:Analysis

16. 3. Recall: Index Shifting for Summations (Use this if…)

17. Index Shifting Sigma representation of a summation is not unique

18. Index Shifting Rules

19. decrease the index by 1 increase the i in the summation by 1

20. Index Shifting Rules increase the index by 1 decrease the i in the summation by 1