1. 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.

2. Example: Binomial coefficients
Example: Use the formula to calculate the combinations , and
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! 50−48 !48!
Modified From Houghton Mifflin Company, Inc.
Example: Binomial coefficients

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

4. 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

5. 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 y10.
Binomial Coefficients
Modified From Houghton Mifflin Company, Inc.

6. 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
2nd row
3rd row
6 + 4 = 10
4th row
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

7. Example: Pascal’s Triangle
Example: Use Pascal’s Triangle to expand (2a + b)4.
1 1
1st row
2nd row
3rd row
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

8. 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
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Definition: Binomial Theorem

9. 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

10. 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 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

11. 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
−1 (−1) 1
=6(32)(−1)
= -192
Modified From Houghton Mifflin Company, Inc.

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