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Warm Up. CST Review. Snowboarding. A sporting goods store offers 3 types of snowboards (all-mountain, freestyle, and carving) and 2 types of boots (soft and hybrid). How many choices does the store offer for snowboarding equipment?. EXAMPLE 1. Use a tree diagram. SOLUTION.

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Presentation Transcript
slide1

Warm Up

CST Review

slide2

Snowboarding

A sporting goods store offers 3 types of snowboards (all-mountain, freestyle, and carving) and 2 types of boots (soft and hybrid). How many choices does the store offer for snowboarding equipment?

EXAMPLE 1

Use a tree diagram

SOLUTION

Draw a tree diagram and count the number of branches.

slide3

ANSWER

The tree has 6 branches. So, there are 6 possible choices.

EXAMPLE 1

Use a tree diagram

slide4

Photography

You are framing a picture. The frames are available in 12 different styles. Each style is available in 55 different colors. You also want blue mat board, which is available in 11 different shades of blue. How many different ways can you frame the picture?

EXAMPLE 2

Use the fundamental counting principle

slide5

ANSWER

The number of different ways you can frame the picture is 7260.

= 7260

= 125511

Number of ways

EXAMPLE 2

Use the fundamental counting principle

SOLUTION

You can use the fundamental counting principle to find the total number of ways to frame the picture. Multiply the number of frame styles (12), the number of frame colors (55), and the number of mat boards (11).

slide6

How many different license plates are possible if letters and digits can be repeated?

How many different license plates are possible if letters and digits cannot be repeated?

EXAMPLE 3

Use the counting principle with repetition

License Plates

The standard configuration for a Texas license plate is 1 letter followed by 2 digits followed by 3 letters.

slide7

There are 26choices for each letter and 10choices for each digit. You can use the fundamental counting principle to find the number of different plates.

= 261010262626

Number of plates

ANSWER

With repetition, the number of different license plates is 45,697,600.

EXAMPLE 3

Use the counting principle with repetition

SOLUTION

= 45,697,600

slide8

If you cannot repeat letters there are still 26choices for the first letter, but then only 25remaining choices for the second letter, 24choices for the third letter, and 23choices for the fourth letter. Similarly, there are 10choices for the first digit and 9choices for the second digit. You can use the fundamental counting principle to find the number of different plates.

= 26109252423

Number of plates

Without repetition, the number of different license plates is 32,292,000.

ANSWER

EXAMPLE 3

Use the counting principle with repetition

= 32,292,000

slide9

SPORTING GOODS The store in Example 1 also offers 3 different types of bicycles (mountain, racing, and BMX) and 3 different wheel sizes (20in.,22 in., and 24in.). How many bicycle choices does the store offer?

ANSWER

9 bicycles

for Examples 1, 2 and 3

GUIDED PRACTICE

slide10

WHAT IF?In Example 3, how do the answers change for the standard configuration of a New York license plate, which is 3 letters followed by 4 numbers?

ANSWER

  • The number of plates would increase to 175,760,000.
  • The number of plates would increase to 78,624,000.

for Examples 1, 2 and 3

GUIDED PRACTICE

slide11

In how many different ways can the bobsledding teams finish the competition? (Assume there are no ties.)

In how many different ways can 3 of the bobsledding teams finish first, second, and third to win the gold, silver, and bronze medals?

EXAMPLE 4

Find the number of permutations

Olympics

Ten teams are competing in the final round of the Olympic four-person bobsledding competition.

slide12

There are 10! different ways that the teams can finish the competition.

= 10 987654 3 2 1

10!

Any of the 10teams can finish first, then any of the remaining 9teams can finish second, and finally any of the remaining 8teams can finish third. So, the number of ways that the teams can win the medals is:

= 720

1098

EXAMPLE 4

Find the number of permutations

SOLUTION

= 3,628,800

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ANSWER

The number of ways to finish would increase to 479,001,600.

The number of ways to finish would increase to 1320.

for Example 4

GUIDED PRACTICE

WHAT IF?In Example 4, how would the answers change if there were 12 bobsledding teams competing in the final round of the competition?

slide14

St 18 & 19

Finding permutations of n objects taken r at a time

n Factorial or n! = n ( n – 1 ) ( n – 2 ) …. 3 * 2 * 1

Examples:

4 ! = 4 * 3 * 2 *1 = 24

8! = 8*7*6*5*4*3*2*1 = 40320

The number of permutations of a set of n objects taken

r at a time (without repetition), denoted by nPr =

n!

( n – r )!

slide15

= 11,880

12!

12!

479,001,600

12P4

=

=

=

8!

40,320

ANSWER

( 12 – 4 )!

You can burn 4 of the 12 songs in 11,880 different orders.

EXAMPLE 5

Find permutations of n objects taken r at a time

Music

You are burning a demo CD for your band. Your band has 12 songs stored on your computer. However, you want to put only 4 songs on the demo CD. In how many orders can you burn 4 of the 12 songs onto the CD?

SOLUTION

Find the number of permutations of 12objects taken 4at a time.

slide16

5P3

4P1

ANSWER

= 60

8P5

= 6720

ANSWER

ANSWER

= 4

for Example 5

GUIDED PRACTICE

Find the number of permutations.

slide17

12P7

ANSWER

= 3,991,680

for Example 5

GUIDED PRACTICE

Find the number of permutations.

Homework: Section 15-1 page 648 problems 2-42

slide18

Find the number of distinguishable permutations of the letters in

MIAMI and

TALLAHASSEE.

MIAMI has 5letters of which M and I are each repeated 2times. So, the number of distinguishable permutations is:

5!

120

=

= 30

2 2

2! 2!

EXAMPLE 6

Find permutations with repetition

SOLUTION

slide19

TALLAHASSEE has 11letters of which A is repeated 3times, and L, S, and E are each repeated 2times. So, the number of distinguishable permutations is:

39,916,800

11!

=

6 2 2 2

3! 2! 2! 2!

EXAMPLE 6

Find permutations with repetition

= 831,600

slide20

MALL

ANSWER

12

for Example 6

GUIDED PRACTICE

Find the number of distinguishable permutations of the letters in the word.

slide21

KAYAK

ANSWER

30

for Example 6

GUIDED PRACTICE

Find the number of distinguishable permutations of the letters in the word.

slide22

ANSWER

50,400

CINCINNATI

for Example 6

GUIDED PRACTICE

Find the number of distinguishable permutations of the letters in the word.

slide23

Permutations with Identical Objects

Homework

Section 15-2 PH Book

Problems #1-20 all