MAT 4830 Numerical Analysis Binomial Coefficients and Combinatorial Identities

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Presentation transcript:

MAT 4830 Numerical Analysis Binomial Coefficients and Combinatorial Identities

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

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

Binomial Expansion

Binomial Theorem

Useful Formulas for Binomial Coefficients

Pascal’s Identity

Proof:Analysis

Binomial Theorem Combinatorial Proof:Analysis

Example 1

Example 2 Proof:Analysis

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

Example 3 (b) Solution:Analysis

Binomial Theorem Induction Proof: Need some preparations Analysis

Binomial Theorem Proof:Analysis

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

Index Shifting Sigma representation of a summation is not unique

Index Shifting Rules

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

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

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