1. Use Combinations and the Binomial Theorem (continued)
Lesson 10.2
Algebra II

2. Agenda Homework Review
Continue discussion about combinations and binomial theorem.
Homework: The rest of homework #3 from the assignment sheet
Quiz on Friday lessons 10.1 – 10.2

3. Use Combinations and the Binomial Theorem
Formally… Combinations of n objects taken r at a time The number of combinations of r objects taken from a group of n distinct objects is denoted by 𝑛 𝐶 𝑟 = 𝑛! 𝑟!(𝑛−𝑟)! DEFINITION SLIDE

4. Use Combinations and the Binomial Theorem
Let’s look at the problem b from the fake quiz… 3 scoops from 25 flavors in a bowl… 25∗24∗23 3∗2∗1 Written as a combination…. 25 𝐶 3 = 25! 3!(25−3)! =2300 EXAMPLE SLIDE

5. Use Combinations and the Binomial Theorem
Do problems 1 and 3 from Practice 10.2 B.

6. Use Combinations and the Binomial Theorem
Sometimes we have multiple events… For example.. Let’s say that we want to know how many different choices we would have to select from for a 2 scoop bowl and a 3 scoop bowl (from 25 flavors..). How would you figure that out? Example Slide

7. Use Combinations and the Binomial Theorem

8. example 2 from the book: William Shakespeare wrote 38 plays that can be divided into three genres; 18 are comedies, 10 are histories and 10 are tragedies. a. How many different sets of exactly 2 comedies and 1 tragedy can you read?

9. example 2 from the book:
William Shakespeare wrote 38 plays that can be divided into three genres; 18 are comedies, 10 are histories and 10 are tragedies.
How many different sets of exactly 2 comedies and 1 tragedy can you read?
i. To get started… how many combinations do you have of 2 comedies?
ii. How many combinations do you have of 1 tragedies?
iii. Can you use the counting principle?

10. example 2 from the book: William Shakespeare wrote 38 plays that can be divided into three genres; 18 are comedies, 10 are histories and 10 are tragedies. b. How many different sets of at most 3 plays can you read? To answer this question we need to determine what is really being asked… at most means that we could read none, one, two or three… so it becomes the sum of each of those…

11. example 2 from the book: William Shakespeare wrote 38 plays that can be divided into three genres; 18 are comedies, 10 are histories and 10 are tragedies. b. How many different sets of at most 3 plays can you read? 38 𝐶 𝐶 𝐶 𝐶 3 = 38! 0!38! + 38! 1!37! !36! !35! = =9178

12. Use Combinations and the Binomial Theorem
Let’s try problems 12 and 13 on 10.2 practice B.

13. Use Combinations and the Binomial Theorem
Yesterday’s lesson focused on Pascal’s triangle and that the numbers in the triangle are the same as the number of combinations. From the homework last night’s homework you saw that when you expand binomials that the coefficients are the same as the numbers in Pascal’s triangle. So, can we use the combinatorial coefficients to expand binomials?

14. Use Combinations and the Binomial Theorem
So (𝑎+𝑏 ) 4 = 𝑎 4 +4 𝑎 3 𝑏+6 𝑎 2 𝑏 2 +4𝑎 𝑏 3 + 𝑏 4

15. Use Combinations and the Binomial Theorem
Binomial Theorem: For any positive integer n, the binomial expansion of (𝑎+𝑏 ) 𝑛 is: (𝑎+𝑏 ) 𝑛 = 𝑛 𝐶 0 𝑎 𝑛 𝑏 0 + 𝑛 𝐶 1 𝑎 𝑛−1 𝑏 + 𝑛 𝐶 2 𝑎 𝑛−2 𝑏 2 +… + 𝑛 𝐶 𝑛 𝑎 0 𝑏 𝑛 DEFINITION SLIDE

16. Use Combinations and the Binomial Theorem
Binomial Theorem: (𝑎+𝑏 ) 𝑛 = 𝑛 𝐶 0 𝑎 𝑛 𝑏 0 + 𝑛 𝐶 1 𝑎 𝑛−1 𝑏 + 𝑛 𝐶 2 𝑎 𝑛−2 𝑏 2 +… + 𝑛 𝐶 𝑛 𝑎 0 𝑏 𝑛 Find: (2𝑥−3 ) 4 = 4 𝐶 0 2𝑥 4 − 𝐶 1 2𝑥 3 (−3) + 4 𝐶 2 (2𝑥) 2 (−3) 𝐶 3 (2𝑥) 1 (−3) 𝐶 4 (2𝑥) 0 (−3) 4 =1∗16 𝑥 4 ∗1+4∗8 𝑥 3 −3 +6∗4 𝑥 2 ∗9 +4∗2𝑥 −27 +1∗(2𝑥 ) 0 (−81) =16 𝑥 4 −96 𝑥 𝑥 2 −216𝑥+81 EXAMPLE SLIDE

17. Use Combinations and the Binomial Theorem
(2𝑥−3 ) 4 =16 𝑥 4 −96 𝑥 𝑥 2 −216𝑥+81 (𝑥+1 ) 3 = 𝑥 3 +3 𝑥 2 +3𝑥+1 (𝑥−1 ) 3 = 𝑥 3 −3 𝑥 2 +3𝑥−1 When you have (𝑎+𝑏 ) 𝑛 what do you notice about the signs? Write a sentence with you answer. When you have (𝑎−𝑏 ) 𝑛 what do you notice about the signs? Write a sentence with you answer. CONCEPT SLIDE

18. Use Combinations and the Binomial Theorem
Do problems 15 and 16 from 10.2, practice B.

19. Use Combinations and the Binomial Theorem
You can also use the binomial theorem to find the coefficient in an expansion. Find the coefficient of 𝑥 3 in the expansion of (𝑥−2 ) 6 … EXAMPLE SLIDE

20. Use Combinations and the Binomial Theorem
Summary of Lesson 10.2:
Major Concepts –
Finding the number of combinations
𝑛 𝐶 𝑟 = 𝑛! 𝑟!(𝑛−𝑟)!
Where n is the number of objects available..(pizza toppings) and r is the number chosen (how many toppings on your pizza.)
SUMMARY SLIDE

21. Use Combinations and the Binomial Theorem
Major Concepts Continued 2. The Binomial Theorem (𝑎+𝑏 ) 𝑛 = 𝑛 𝐶 0 𝑎 𝑛 𝑏 0 + 𝑛 𝐶 1 𝑎 𝑛−1 𝑏 + 𝑛 𝐶 2 𝑎 𝑛−2 𝑏 2 +… + 𝑛 𝐶 𝑛 𝑎 0 𝑏 𝑛 The number of combinations can be used for expanding a binomial. SUMMARY SLIDE

22. Use Combinations and the Binomial Theorem
Major Concepts Continued: 3. Pascal’s triangle Both the number of combinations and the coefficients for binomial expansions can be found in Pascal’s triangle. SUMMARY SLIDE