Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved

Chapter 11

Counting Methods

© 2008 Pearson Addison-Wesley.All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved

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Chapter 11: Counting Methods

11.1 Counting by Systematic Listing

11.2 Using the Fundamental Counting Principle

11.3 Using Permutations and Combinations

11.4 Using Pascal’s Triangle

11.5 Counting Problems Involving “Not” and

“Or”


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Chapter 1

Section 11-2Using the Fundamental Counting

Principle


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Using the Fundamental Counting Principle

• Uniformity and the Fundamental Counting Principle

• Factorials

• Arrangements of Objects


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Uniformity Criterion for Multiple-Part Tasks

A multiple-part task is said to satisfy the uniformity criterion if the number of choices for any particular part is the same no matter which choices were selected for the previous parts.


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Fundamental Counting Principle

When a task consists of k separate parts and satisfies the uniformity criterion, if the first part can be done in n1 ways, the second part can be done in n2 ways, and so on through the k th part, which can be done in nk ways, then the total number of ways to complete the task is given by the product

1 2 3 .kn n n n


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Example: Two-Digit Numbers

How many two-digit numbers can be made from the set {0, 1, 2, 3, 4, 5}? (numbers can’t start with 0.)

Solution

Part of Task Select first digit Select second digit

Number of ways 5

(0 can’t be used)

6

There are 5(6) = 30 two-digit numbers.


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Example: Two-Digit Numbers with Restrictions

How many two-digit numbers that do not contain repeated digits can be made from the set {0, 1, 2, 3, 4, 5} ?

SolutionPart of Task

Select first digit

Select second digit

Number of ways

5 5

(repeated digits not allowed)

There are 5(5) = 25 two-digit numbers.


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Example: Two-Digit Numbers with Restrictions

How many ways can you select two letters followed by three digits for an ID?

SolutionPart of Task

First letter

Second letter

Digit Digit Digit

Number of ways

26 26 10 10 10

There are 26(26)(10)(10)(10) = 676,000 IDs possible.


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Factorials

For any counting number n, the product of all counting numbers from n down through 1 is called n factorial, and is denoted n!.


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Factorial Formula

For any counting number n, the quantity n factorial is given by

! ( 1)( 2) 2 1.n n n n


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Example:

Evaluate each expression.

a) 4! b) (4 – 1)! c)5!

3!Solution

a) 4! 4 3 2 1 24

b) (4 1)! 3 2 1 6

5! 5 4 3!c) 5 4 20

3! 3!


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Definition of Zero Factorial

0! 1


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Arrangements of Objects

When finding the total number of ways to arrange a given number of distinct objects, we can use a factorial.


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Arrangements of n Distinct Objects

The total number of different ways to arrange n distinct objects is n!.


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Example: Arranging Books

How many ways can you line up 6 different books on a shelf?

Solution

The number of ways to arrange 6 distinct objects is 6! = 720.


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Arrangements of n Objects Containing Look-Alikes

The number of distinguishable arrangements of n objects, where one or more subsets consist of look-alikes (say n1 are of one kind, n2 are of another kind, …, and nk are of yet another kind), is given by

1 2

!.

! ! !k

n

n n n


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Example: Distinguishable Arrangements

Determine the number of distinguishable arrangements of the letters of the word INITIALLY.

Solution

9!30240.

3!2!9 letters total

3 I’s and 2 L’s

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Chapter 11 Counting Methods © 2008 Pearson Addison-Wesley. All rights reserved