copyright 2013, 2010, 2007, pearson, education, inc. section 12.8 the counting principle and...

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 12.8

The Counting Principle

and Permutations


What You Will Learn

The Counting PrinciplePermutations

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Counting Principle

If a first experiment can be performed in M distinct ways and a second experiment can be performed in N distinct ways, then the two experiments in that specific order can be performed in M • N distinct ways.

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Example 1: Counting Principle: PasswordsA password used to gain access to a computer account is to consist of two lower case letters followed by four digits. Determine how many different passwords are possible if a) repetition of letters and digits is permitted.Solution26•26•10•10•10•10 = 6,760,000 different possible passwords

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Example 1: Counting Principle: Passwordsb) repetition of letters and digits is not permitted.

Solution

26•25•10•9•8•7 = 3,276,000 different possible passwords

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Example 1: Counting Principle: Passwordsc) the first letter must be a vowel (a, e, i, o, u) and the first digit cannot be a 0, and repetition of letters and digits is not permitted.

Solution5•25•9•9•8•7 = 567,000 different possible passwords

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Permutations

A permutation is any ordered arrangement of a given set of objects.

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Number of Permutations

The number of permutations of n distinct items is n factorial, symbolized n!, where

n! = n(n – 1)(n – 2) • • • (3)(2)(1)

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Example 3: Cell Phones

In how many different ways can six different cell phones be arranged on top of one another?Solution

6! = 6 • 5 • 4 • 3 • 2 • 1 = 720The 6 cell phones can be arranged in 720 different ways.

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Example 4: Permutation of Three Out of Five LettersConsider the five letters a, b, c, d, e. In how many distinct ways can three letters be selected and arranged if repetition is not allowed?Solution

5 • 4 • 3 = 60Thus, there are 60 different possible ordered arrangements, or permutations.

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Permutation Formula

The number of permutations possible when r objects are selected from n objects is found by the permutation formula

nP

r

n!

n r !12.8-11


Example 5: Using the Permutation FormulaYou are among eight people forming a skiing club. Collectively, you decide to put each person’s name in a hat and to randomly select a president, a vice president, and a secretary. How many different arrangements or permutations of officers are possible?

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Example 5: Using the Permutation FormulaSolutionn = 8, r = 3

8P

3

8!

8 3 ! 8!

5!

8 7 6 5 4 32 1

5 4 32 1 336

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Permutations of Duplicate ObjectsThe number of distinct permutations of n objects where

n1 of the objects are identical,

n2 of the objects are identical,

…, nr of the objects are

identical is found by the formula

n!

n1!n

2!n

r!

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Example 7: Duplicate LettersIn how many different ways can the letters of the word “TALLAHASSEE” be arranged?SolutionOf the 11 letters, 3 are A’s, 2 are S’s, 2 are L’s, and 2 are E’s.

11!

3!2!2!2!

1110 9 8 7 6 5 42

32 1

32 1 32 12 1

831,60012.8-15

copyright 2013, 2010, 2007, pearson, education, inc. section 12.8 the counting principle and...

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