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Slide 11.6- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 1 2 3 4
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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.
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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 101010101010101010 = 10 9 There are 1,000,000,000 possible Social Security numbers.
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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.
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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 That is, n distinct objects can be arranged in n! different ways.
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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
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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. b.
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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.
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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:
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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. There are 56 ways to select 5 of 8 pizza toppings.
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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 n 1 are of one kind, n 2 are of second kind, …, and n k are of a kth kind is where n 1 + n 1 + + n k = n.
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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. There are 1680 ways the nine gifts can be distributed among the three children.
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