Use Combinations and the Binomial Theorem (continued)

Lesson 10.2Algebra II

Agenda

1. Homework Review2. Continue discussion about combinations and

binomial theorem.3. Homework: The rest of homework #3 from

the assignment sheet4. Quiz on Friday lessons 10.1 – 10.2

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

DEFINITION SLIDE

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…

Written as a combination….

EXAMPLE SLIDE

Use Combinations and the Binomial Theorem

Do problems 1 and 3 from Practice 10.2 B.

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

Use Combinations and the Binomial Theorem

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?

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?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?

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…

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?

Use Combinations and the Binomial Theorem

Let’s try problems 12 and 13 on 10.2 practice B.

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?

Use Combinations and the Binomial Theorem

So..

Use Combinations and the Binomial Theorem

Binomial Theorem:

For any positive integer n, the binomial expansion of is:

DEFINITION SLIDE

Use Combinations and the Binomial Theorem

Binomial Theorem:

Find:

EXAMPLE SLIDE

Use Combinations and the Binomial Theorem

When you have what do you notice about the signs? Write a sentence with you answer.When you have what do you notice about the signs? Write a sentence with you answer.CONCEPT SLIDE

Use Combinations and the Binomial Theorem

Do problems 15 and 16 from 10.2, practice B.

Use Combinations and the Binomial Theorem

You can also use the binomial theorem to find the coefficient in an expansion.

Find the coefficient of in the expansion of …

EXAMPLE SLIDE

Use Combinations and the Binomial Theorem

Summary of Lesson 10.2:Major Concepts – 1. Finding the number of combinations

Where n is the number of objects available..(pizza toppings) and r is the number chosen (how many toppings on your pizza.)SUMMARY SLIDE

Use Combinations and the Binomial Theorem

Major Concepts Continued2. The Binomial Theorem

The number of combinations can be used for expanding a binomial.

SUMMARY SLIDE

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