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Understanding Binomial Coefficients and Combinatorial Identities in Numerical Analysis

This resource covers essential aspects of binomial coefficients and combinatorial identities, vital for understanding the Binomial Theorem. It includes a thorough review of index shifting methods, provides examples of binomial expansions, and offers useful formulas such as Pascal’s Identity. Additionally, it features a take-home exam to reinforce learning. This material is designed for students seeking to enhance their comprehension of numerical analysis concepts related to combinatorial mathematics.

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Understanding Binomial Coefficients and Combinatorial Identities in Numerical Analysis

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  1. MAT 4830Numerical 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. Pascal’s Identity

  9. Binomial Theorem

  10. Example 1 Find the coefficient of in the expansion of .

  11. Example 2

  12. Example 3 (a)

  13. Example 3 (b)

  14. Binomial Theorem

  15. Binomial Theorem

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

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

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