The Binomial Theorem 9-5
Combinations How many combinations can be created choosing r items from n choices. 4! = (4)(3)(2)(1) = 24 0! = 1 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
Combinations If there are 4 toppings to choose from and I can afford a 2 topping pizza how many possible pizzas do I have to choose from? Toppings: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Pepperoni Artichokes Olives Sardines
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Binomial Expansions Consider the patterns formed by expanding (x + y) n. (x + y) 0 = 1 (x + y) 1 = x + y (x + y) 2 = x 2 + 2xy + y 2 (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3 (x + y) 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 (x + y) 5 = x 5 + 5x 4 y + 10x 3 y x 2 y 3 + 5xy 4 + y 5 Notice that each expansion has n + 1 terms. 1 term 2 terms 3 terms 4 terms 5 terms 6 terms Example: (x + y) 10 will have , or 11 terms.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Patterns of Exponents in Binomial Expansions Consider the patterns formed by expanding (x + y) n. (x + y) 0 = 1 (x + y) 1 = x + y (x + y) 2 = x 2 + 2xy + y 2 (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3 (x + y) 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 (x + y) 5 = x 5 + 5x 4 y + 10x 3 y x 2 y 3 + 5xy 4 + y 5 1. The exponents on x decrease from n to 0. The exponents on y increase from 0 to n. 2. Each term is of degree n. Example: The 5 th term of (x + y) 10 is a term with x 6 y 4.”
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Binomial Coefficients The coefficients of the binomial expansion are called binomial coefficients. The coefficients have symmetry. (x + y) 5 = x 5 + 5x 4 y + 10x 3 y x 2 y 3 + 5xy 4 + y 5 The first and last coefficients are 1. The coefficients of the second and second to last terms are equal to n. 1 1 Example: What are the last 2 terms of (x + y) 10 ? Since n = 10, the last two terms are 10xy 9 + 1y 10.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 The Binomial Theorem! Definition: Binomial Theorem r is defined as 1 less than the term number The coefficient of x n–r y r in the expansion of (x + y) n is written or n C r. Example: What are the last 2 terms of (x + y) 10 ? Since n = 10, the last two terms are 10xy 9 + 1y 10. So, the last two terms of (x + y) 10 can be expressed as 10 C 9 xy C 10 y 10 or as xy 9 + y 10. 0! is defined to be 1.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 The Binomial Theorem! Example 1: Use the Binomial Theorem to expand (x 4 + 2) 3. Definition: Binomial Theorem r is defined as 1 less than the term number Easier way? You know it!
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Pascal’s Triangle The triangular arrangement of numbers below is called Pascal’s Triangle. Each number in the interior of the triangle is the sum of the two numbers immediately above it. The numbers in the n th row of Pascal’s Triangle are the binomial coefficients for (x + y) n. 1 1 st row nd row rd row th row th row 0 th row = = 3
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example 2: Use the fifth row of Pascal’s Triangle to generate the sixth row and find the binomial coefficients, 5 th row th row 6 C 0 6 C 1 6 C 2 6 C 3 6 C 4 6 C 5 6 C 6 There is symmetry between binomial coefficients. n C r = n C n–r Example: Pascal’s Triangle
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example 4: Use Pascal’s Triangle to expand (2a + b) 4. (2a + b) 4 = 1(2a) 4 + 4(2a) 3 b + 6(2a) 2 b 2 + 4(2a)b 3 + 1b 4 = 1(16a 4 ) + 4(8a 3 )b + 6(4a 2 b 2 ) + 4(2a)b 3 + b 4 = 16a a 3 b + 24a 2 b 2 + 8ab 3 + b 4 Example: Pascal’s Triangle 1 1 st row nd row rd row th row 0 th row1
Ex 5 Find the binomial coefficients of a binomial expansion raised to the 6 th power. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12