Slide 11.6- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Counting Principles

Learn to use the Fundamental Counting Principle.Learn to use the formula for permutations.Learn to use the formula for combinations.Learn to use the formula for distinguishable permutations.

SECTION 11.6

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3

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Slide 11.6- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

FUNDAMENTAL COUNTING PRINCIPLE

If a first choice can be made in p different ways, a second choice can be made in q different ways, a third choice can be made in r different ways, and so on, then the sequence of choices can be made in p • q • r • • • different ways.

Slide 11.6- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 3 Counting Possible Social Security Numbers

Social Security numbers have the format NNN-NN-NNNN, where each N must be one of the integers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Assuming that there are no other restrictions, how many such numbers are possible?

Solution

There are 10 possible digits for each position. NNN-NN-NNNN

10•10•10•10•10•10•10•10•10 = 109

There are 1,000,000,000 possible Social Security numbers.

Slide 11.6- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

DEFINITION OF PERMUTATION

A permutation is an arrangement of n distinct objects in a fixed order in which no object is used more than once. The specific order is important: Each different ordering of the same objects is a different permutation.

Slide 11.6- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

NUMBER OF PERMUTATIONS OF n OBJECTS

The number of permutations of n distinct objects is

n!n n 1 L L 4 321.

That is, n distinct objects can be arranged in n! different ways.

Slide 11.6- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

PERMUTATIONS OF n OBJECTS TAKEN r AT A TIME

The number of permutations of n distinct objects taken r at a time is denoted by P(n, r), where

P n,r n!

n r !n n 1 n 2 L n r 1 .

Slide 11.6- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 6 Using the Permutation Formula

Use the formula for P(n, r) to evaluate each expression.a. P(7, 3) b. P(6, 0)

Solution

a. P 7, 3 7!

7 3 ! 7!

4!

7654!

4!210

b. P 6,0 6!

6 0 ! 6!

6!1

Slide 11.6- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

DEFINITION OF A COMBINATION OF n DISTINCT OBJECTS TAKEN r AT A TIME

When r objects are chosen from n distinct objects without regard to order, we call the set of r objects a combination of n objects taken r at a time.The symbol C(n, r) denotes the total number of combinations of n objects taken r at a time.

Slide 11.6- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

THE NUMBER OF COMBINATIONS OF n DISTINCT OBJECTS TAKEN r AT A TIME

The number of combinations of n distinct objects taken r at a time is:

C n,r n!

n r !r!

Slide 11.6- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 9 Choosing Pizza Toppings

How many different ways can five pizza toppings be chosen from the following choices: pepperoni, onions, mushrooms, green peppers, olives, tomatoes, mozzarella, and anchovies?Solution

We need to find the number of combinations of 8 objects taken 5 at a time.

C 8,5 8!

8 5 !5!

8!

3!5!

876321

56

There are 56 ways to select 5 of 8 pizza toppings.

Slide 11.6- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

DISTINGUISHABLE PERMUTATIONS

The number of distinguishable permutations of n objects of which n1 are of one kind, n2 are of second kind, …, and nk are of a kth kind is

n!

n1!n2 !L L nk !

where n1 + n1 + • • • + nk = n.

Slide 11.6- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

EXAMPLE 10 Distributing Gifts

In how many ways can nine gifts be distributed among three children if each child receives three gifts?

Solution

Count the number of permutations of 9 objects, of which three are of one kind, three are of a second kind and three are of a third kind.

9!

3!3!3!1680

There are 1680 ways the nine gifts can be distributed among the three children.