Sec 4.4

Counting Rules

Bluman, Chapter 4

A Question to Ponder:

A box contains 3 red chips, 2 blue chips and 5 green chips. A chip is selected, replaced and a second chip is selected.

Display the sample space.

Do you think each event is equally likely?

A Question to Ponder:

A box contains 3 red chips, 2 blue chips and 5 green chips. A chip is selected, replaced and a second chip is selected. Find the following probabilities

a) Selecting 1 red chip and then 1 blue.b) Selecting 1 blue chip and then 1 green chip.c) Selecting 2 blue chips.

A Question to Ponder:

A box contains 3 red chips, 2 blue chips and 5 green chips. A chip is selected, and not replaced and a second chip is selected.

Display the sample space

Do you think each event is equally likely?

A Question to Ponder:

A box contains 3 red chips, 2 blue chips and 5 green chips. A chip is selected, and not replaced and a second chip is selected. Find the probability of each event

a) Selecting 1 red chip and then 1 blue chip.b) Selecting 1 blue chip and then 1 green chip.c) Matching chips.

Another Question to Ponder

A game is played by drawing four cards from an ordinary deck and replacing each card after it is drawn. Find the probability of winning if at least one ace is drawn.

The fundamental counting rule is also called the multiplication of choices.

In a sequence of n events in which the first one has k1 possibilities and the second event has k2 and the third has k3, and so forth, the total number of possibilities of the sequence will be

k1 · k2 · k3 · · · kn

4.4 Counting Rules

Bluman, Chapter 4

Chapter 4Probability and Counting Rules

Section 4-4Example 4-39

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Example 4-39: Paint ColorsA paint manufacturer wishes to manufacture several different paints. The categories include

Color: red, blue, white, black, green, brown, yellow

Type: latex, oil

Texture: flat, semigloss, high gloss

Use: outdoor, indoor

How many different kinds of paint can be made if you can select one color, one type, one texture, and one use?

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# of # of # of # ofcolors types textures uses

7 2 3 2

84 different kinds of paint

Example 4-40 There are four blood types, A, B, AB, and

O. Blood can also be Rh+ and Rh-. Finally, a blood donor can be classified as either male or female. How many different ways can a donor have his or her blood labeled?Blood type ∙sign ∙gender

4∙2∙2=16

Example 4-41 The digits 1, 2, 3, 4, 5 and 6 are to be

used in a four-digit ID card. How many different cards are possible if repetitions are permitted?

First digit ∙ second digit ∙ third ∙ fourth

6 ∙ 6 ∙ 6 ∙ 6= 1296

Counting Rules Factorial is the product of all the

positive numbers from 1 to a number.

Bluman, Chapter 4

! 1 2 3 2 1

0! 1

n n n n

While you wait:

In how many ways can 3 people stand in line at a store?

In how many ways can 4 people be chosen from a class of 12 to go on a FT?

Bluman, Chapter 4

Review topicsIn how many ways can two items be chosen from a selection of 4 . No replacement.

Bluman, Chapter 4

Consider the following cases

Ten runners compete in an event.

The top 3 finishers advance to the State level.

Ten runners compete in an event.

The first to cross the finish line will receive a gold medal, the second place a silver and finally the third finisher a bronze medal.

Bluman, Chapter 4

Bluman, Chapter 4

There is 5-question True or False test: T=true; F=false

Are the following the same result:

TFFTT OR FFTTT

There are 5 DVDs. Marge is sick and staying home. In how many ways can Marge watch the DVDs.

Bluman, Chapter 4

Bluman, Chapter 4

Marge is shopping at the local grocery store and needs five items from the produce department. Jack is also buying the same five items.

Do you think Jack will make his choices in the same order as Marge?

Does it matter at the end how they make their choices?

There are 8 DVDs. Marge is sick and staying home. In how many ways can Marge watch the DVDs?

a) State when order matters.

b) State when order doesn’t matter.

Bluman, Chapter 4

Permutation Rule The arrangement of n objects in a specific order

using r objects at a time is called a permutation of n objects taking r objects at a time. It is

written as nPr and the formula is

n rpn

n r

!

( )!

Counting Rules Combination is a grouping of objects. Order does not matter.

Bluman, Chapter 4

!

! !n r

nC

n r r

!n rP

r

Permutation and combination

Order matters:

nPr

Order does NOT matter

nCr

Bluman, Chapter 4

Chapter 4Probability and Counting Rules

Section 4-4Example 4-42/4-43

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Example 4-42: Business LocationsSuppose a business owner has a choice of 5 locations in which to establish her business. She decides to rank each location according to certain criteria, such as price of the store and parking facilities. How many different ways can she rank the 5 locations?

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first second third fourth fifthchoice choice choice choice choice

120 different ways to rank the locations

5 4 3 2 1

Using factorials, 5! = 120.

Using permutations, 5P5 = 120.

Example 4-43: Business LocationsSuppose the business owner in Example 4–42 wishes to rank only the top 3 of the 5 locations. How many different ways can she rank them?

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first second thirdchoice choice choice

60 different ways to rank the locations

5 4 3

Using permutations, 5P3 = 60.

Chapter 4Probability and Counting Rules

Section 4-4Example 4-44

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Example 4-44: Television News StoriesA television news director wishes to use 3 news stories on an evening show. One story will be the lead story, one will be the second story, and the last will be a closing story. If the director has a total of 8 stories to choose from, how many possible ways can the program be set up?

Since there is a lead, second, and closing story, we know that order matters. We will use permutations.

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8 3

8!336

5!P 8 3

3

or 8 7 6 336P

Chapter 4Probability and Counting Rules

Section 4-4Example 4-45

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Example 4-45: School Musical PlaysA school musical director can select 2 musical plays to present next year. One will be presented in the fall, and one will be presented in the spring. If she has 9 to pick from, how many different possibilities are there?

Order matters, so we will use permutations.

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9 2

9!72

7!P 9 2

2

or 9 8 72P

Chapter 4Probability and Counting Rules

Section 4-4Example 4-48

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Example 4-48: School MusicalsA newspaper editor has received 8 books to review. He decides that he can use 3 reviews in his newspaper. How many different ways can these 3 reviews be selected?

The placement in the newspaper is not mentioned, so order does not matter. We will use combinations.

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8 3

8!8!/ 5!3! 56

5!3!C

8 3

8 7 6or 56

3 2C

8 3

8 3or 563!

PC

Chapter 4Probability and Counting Rules

Section 4-4Example 4-49

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Example 4-49: Committee SelectionIn a club there are 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there?

There are not separate roles listed for each committee member, so order does not matter. We will use combinations.

There are 35·10=350 different possibilities.

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7 3 5 2

7! 5!Women: 35, Men: 10

4!3! 3!2!C C

On your own:

Read: TI 83 84 activities on

pages 208 & 235 Summary Table on

page 232

Answer: Applying Concepts 4-

4 on page 232 Exercises 4-4 starting

with # 1 every other odd problem.

Bluman, Chapter 4