Lesson 12.5, For use with pages 675-679

1. Choose one of 4 friends, one of 4 restaurants, and one of 3 meeting times.

Use the counting principle to find the number of possible choices.

You NEED a scientific calculator TODAY!

2. How many seven digit phone numbers are possible if the first digit cannot be a 0?

Lesson 12.5, For use with pages 675-679

1. Choose one of 4 friends, one of 4 restaurants, and one of 3 meeting times.

ANSWER 48

Use the counting principle to find the number of possible choices.

You NEED a scientific calculator TODAY!

2. How many seven digit phone numbers are possible if the first digit cannot be a 0?

ANSWER 9 x 10 x 10 x 10 x 10 x 10 x 10 = 9,000,000

Permutations(Arrangements)

Section 12.5

P. 675

Essential Questions

• What are the differences between permutations and combinations?

• What are the differences between odds and probability?

• How is probability used to make predictions?

• What are the differences between experimental and theoretical probabilities?

• When looking at options or choices, we often look at the arrangement, or order, of people, letters, numbers, etc.

A permutation is an arrangement in which ORDER is important.

Arranging “ABC” is different than “ACB”

• Let’s look at the three letters – A, B, C

• How many different ways can I arrange or order these letters?

• What if I add “D” to the list? Now, how many different arrangements?

• Let’s look at the three letters – A, B, C

• How many different ways can I arrange or order these letters?

• What if I add “D” to the list? Now, how many different arrangements?

6 ways

24 ways

EXAMPLE 1 Counting Permutations

Music

You have five CDs. You can use the counting principle to count the number of permutations of 5 CDs. This is the number of different orders in which you can listen to the CDs.

Choices for 1st CD

Choices for 2nd CD

Choices for 3rd CD

Choices for 4th CD

Choices for 5th CD

5 4 3 2 1

ANSWER

You can listen to the CDs in 120 different orders.

= 120

Swimming Event• There are 8 lanes for the swimming event. If

the top 2 swimmers are assigned to lanes 3, and 4 in how many ways can the other 6 swimmers be assigned to their lanes?

• (Hint: -- you are only looking at 6 lanes)• For the first of the remaining 6 swimmers, there are 6

possible lanes assignments. Once a selection has been made, there are only 5 lanes left from which to assign the next swimmer. After the second swimmer is assigned, there are only 4 lanes left, and so on.

• 6 x 5 x 4 x 3 x 2 x 1 = 720 ways

• Factorial Notation is used to write the product when the factors are consecutive whole numbers.

• 4 factorial is written as 4! and it means

4 x 3 x 2 x 1 or 24

Find the factorial key on your calculator.

GUIDED PRACTICE for Examples 1 and 2

1. Evaluate 3!.

3! = 3 2 1

2. Evaluate 6!.

6! = 6 5 4 3 2 1

3. Evaluate 1!.

1!

= 720

= 6

= 1

• How many ways can the letters in these words be arranged?

MATH PROBABILITY

GEOMETRY VOLUME

4! = 24 11! = 39,916,800

8! = 40,320 6! = 720

Permutations

n = total to select from

r = the number you are selecting

EXAMPLE 3 Evaluating a Permutation

Poetry

Two students are chosen from a group of 6 to read the first and second poems at the school’s poetry reading. To find how many different ways the students can be chosen, find 6P2.

6P2– == 6!

(6 2)!6!4!

Use permutation formula.

= 6 5 4 3 2 14 3 2 1

Divide out common factors.

= 30 Multiply.

ANSWERThe students can be chosen in 30 ways.Using the counting principle we would have found it by taking 6 x 5 = 30

• From the 9 players in the batting order, in how many ways can a baseball manager choose and order

the first 3 batters?• 9 x 8 x 7 = 504 or 9P3 = 504• the first 4 batters?• 3024 ways the first 5 batters? 15,120 ways

• There are 10 candidates for student government. There are 4 different positions. In how many ways can the student government be chosen?

• 10P4 = 5040 ways

• Remember: these examples all involve ORDER – arrangements / permutations

On Line Calculators

• Permutations and Combinations– http://www.calctool.org/CALC/math/

probability/combinations– http://www.mathsisfun.com/combinatorics/

combinations-permutations-calculator.html

• Factorials– http://joemath.com/math124/Calculator/

factorial.htm

• www.lewiscentral.k12.ia.us/shipp

• Assignment: P. 677 #1-21