the-binomial-theorem.pdf - Unit 2 Mathematical Induction The Binomial Theorem Subtopics I The Pascal\u2019s Triangle II The Concept of Combination III The

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Unit 2 Mathematical Induction
The Binomial Theorem I. The Pascal’s Triangle II. The Concept of Combination III. The Binomial Theorem Subtopics:
The Pascal’s Triangle
Consider the following powers of (a + b): (a + b) 1 = a + b (a + b) 2 = a 2 + 2 ab + b 2 (a + b) 3 = a 3 + 3 a 2 b + 3 ab 2 + b 3 (a + b) 4 = a 4 + 4 a 3 b + 6 a 2 b 2 + 4 ab 3 + b 4 (a + b) 5 = a 5 + 5 a 4 b + 10 a 3 b 2 + 10 a 2 b 3 + 5 ab 4 + b 5 (a + b) 0 = 1 The Pascal’s Triangle
The numerical coefficients of the powers of (a + b) or the Binomial Coefficients : (a + b) 1 = 1 1 (a + b) 2 = 1 2 1 (a + b) 3 = 1 3 3 1 (a + b) 4 = 1 4 6 4 1 (a + b) 5 = 1 5 10 10 5 1 (a + b) 0 = 1 The Pascal’s Triangle
Ex 1. Use the Pascal’s Triangle to expand (2x + 3y) 5 . (2x + 3y) 5 =32x 5 + 240x 4 y + 720x 3 y 2 + 1080x 2 y 3 + 810xy 4 + 243y 5 The Pascal’s Triangle
Ex 2. Use the Pascal’s Triangle to expand (x 2 2y 5 ) 4 . (x 2 2y 5 ) 4 = x 8 8x 6 y 5 + 24x 4 y 10 32x 2 y 15 + 16y 20 The Pascal’s Triangle
The Concept of Combination
The Concept of Combination For any nonnegative integers n and r where n r, the Binomial coefficient , denoted by 𝒏 𝒓 , is defined as 𝒏 𝒓 = 𝒏! 𝒓! 𝒏−𝒓 ! .
? ? = 1 ? ? = 1 ? ? = 1