Combination 𝑛 𝑟 a new mathematical operation! Formula GDC 𝑛 𝑟 = 𝑛! 𝑟! 𝑛−𝑟 ! where 𝑘!=1×2×3×…×𝑘 read “k factorial” Special Factorial –> 0! =1 Enter the value of n MATH, PRB, 3:nCr, ENTER Enter the value of r Modified from Houghton Mifflin Company, Inc.

Example: Binomial coefficients Example: Use the formula to calculate the combinations 10 6 , 13 0 and 50 48 . 10 6 = 10! 10−6 !6! = 10! 4 !6! = 10⋅9∙8∙7 6! 4!6! = 10⋅9∙8∙7 4∙3∙2∙1 =210 13 0 = 13! 13−0 !0! = 13! 13!0! 50 48 = 50! 50−48 !48! Modified From Houghton Mifflin Company, Inc. Example: Binomial coefficients

modified from Houghton Mifflin Company, Inc. Expansion Worksheet Recall - 𝑎+𝑏 3 ≠ 𝑎 3 + 𝑏 3 modified from Houghton Mifflin Company, Inc.

Patterns of Exponents in Binomial Expansions Consider the patterns formed by expanding (a + b)n. (a + b)0 = 1 1 term (a + b)1 = a + b 2 terms (a + b)2 = a2 + 2ab + b2 3 terms (a + b)3 = a3 + 3a2b + 3ab2 + b3 4 terms (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 5 terms (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5 6 terms 1. The exponents on a decrease from n to 0. The exponents on b increase from 0 to n. 2. Each term is of degree n. Example: The 5th term of (a + b)10 is a term with a6b4.” 3. Notice that each expansion has n + 1 terms. Modified from Houghton Mifflin Company, Inc. Patterns of Exponents in Binomial Expansions

Binomial Coefficients The coefficients of the binomial expansion are called binomial coefficients. The coefficients have symmetry. (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 1 The first and last coefficients are 1. The coefficients of the second and second to last terms are equal to n. The coefficient of xn–ryr in the expansion of (x + y)n is written or nCr . So, the last two terms of (x + y)10 can be expressed as 10C9 xy9 + 10C10 y10 or as xy 9 + y10. Binomial Coefficients Modified From Houghton Mifflin Company, Inc.

Modified From Houghton Mifflin Company, Inc. The triangular arrangement of numbers below is called Pascal’s Triangle. 0th row 1 1 1 1st row 1 + 2 = 3 1 2 1 2nd row 1 3 3 1 3rd row 6 + 4 = 10 1 4 6 4 1 4th row 1 5 10 10 5 1 5th row Each number in the interior of the triangle is the sum of the two numbers immediately above it. The numbers in the nth row of Pascal’s Triangle are the binomial coefficients for (x + y)n . Modified From Houghton Mifflin Company, Inc. Pascal’s Triangle

Example: Pascal’s Triangle Example: Use Pascal’s Triangle to expand (2a + b)4. 1 1 1st row 1 2 1 2nd row 1 3 3 1 3rd row 1 4 6 4 1 4th row 0th row 1 (2a + b)4 = 1(2a)4 + 4(2a)3b + 6(2a)2b2 + 4(2a)b3 + 1b4 = 1(16a4) + 4(8a3)b + 6(4a2b2) + 4(2a)b3 + b4 = 16a4 + 32a3b + 24a2b2 + 8ab3 + b4 Modified From Houghton Mifflin Company, Inc. Example: Pascal’s Triangle

Definition: Binomial Theorem general term Binomial Theorem (𝑎+𝑏) 𝑛 = 𝑎 𝑛 + 𝑛 1 𝑎 𝑛−1 𝑏+…+ 𝑛 𝑟 𝑎 𝑛−𝑟 𝑏 𝑟 +…+ 𝑏 𝑛 Example: Use the Binomial Theorem to expand (x4 + 2)3. a b 3 1 3 2 Modified from Houghton Mifflin Company, Inc. Definition: Binomial Theorem

Definition: Binomial Theorem Although the Binomial Theorem is stated for a binomial which is a sum of terms, it can also be used to expand a difference of terms. Simply rewrite (x - y) n as (x + (– y)) n and apply the theorem to this sum. Example: Use the Binomial Theorem to expand (3x – 4)4. Modified From Houghton Mifflin Company, Inc. Definition: Binomial Theorem

Example: Find the nth term Example: Find the eighth term in the expansion of (x + y)13 . Think of the first term of the expansion as x13y 0 . The power of y is 1 less than the number of the term in the expansion. The eighth term is 13 7 x 6 y7. 13 7 Therefore, the eighth term of (x + y)13 is 1716 x 6 y7. Modified From Houghton Mifflin Company, Inc. Example: Find the nth term

Example: Find the coefficient of 𝑏 9 in the expansion of 2 𝑏 2 − 1 𝑏 6 General term: 6 𝑟 2 𝑏 2 6−𝑟 − 1 𝑏 𝑟 = 6 𝑟 2 6−𝑟 𝑏 12−2𝑟 (−1) 𝑟 𝑏 −𝑟 = 6 𝑟 2 6−𝑟 (−1) 𝑟 𝑏 12−3𝑟 12−3𝑟=9 𝑟=1 6 1 2 6−1 (−1) 1 =6(32)(−1) = -192 Modified From Houghton Mifflin Company, Inc.

Skill Practice Pearson Exercise 3.5 #1 a, d, f, g #3 b, c, e #4