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# Section 12.3 - PowerPoint PPT Presentation

Section 12.3. Permutations and Combinations. Permutations. Objectives. Use the Fundamental Counting Principle to count permutations. Evaluate factorial expressions. Use the permutation formula. Find the number of permutations of duplicate items. Key Term:. Permutation:

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### Section 12.3

Permutations and Combinations

### Permutations

• Use the Fundamental Counting Principle to count permutations.

• Evaluate factorial expressions.

• Use the permutation formula.

• Find the number of permutations of duplicate items.

• Permutation:

• An ordered arrangement for items that occurs when

• No item is used more than once.

• The order or arrangement makes a difference.

• When we select r different objects from a set of n objects, symbolized by P(n, r).

• Factorial Notation – If n is a positive integer, the notation n! (read “n factorial”) is the product of all positive integers from n down through 1.

• n! = n(n – 1)(n – 2)…(3)(2)(1)

• 0! by definition, is 1. 0! = 1

• Use the Fundamental Counting Principle

• Five singers are to perform on a weekend evening at a night club. How many different ways are there to schedule their appearances?

• In how many ways can a police department arrange eight suspects in a police lineup if each contains all eight people?

• As in Example 1, five singers are to perform at a night club. One of the singers insists on being the last performer of the evening. If this singer’s request is granted on how many different ways are there to schedule the appearances?

• Evaluate the factorial expression.

• 12!10!

• Evaluate the factorial expression.

• 29!25!

• Evaluate the factorial expression.

• 6! – 3!

• Evaluate the factorial expression.

• (6 – 3)!

• Evaluate the factorial expression.

• Evaluate the factorial expression.

Key Concept:Formula for Computing P(n, r)

• Permutations of “n” things taken “r” at a time – the number of possible permutations if “r” items are taken from “n” items is:

• A corporation has seven members on its board of directors. In how many different ways can it elect a president, vice-president, secretary and treasurer?

• Suppose you asked to list, in order of preference, the three best movies you have seen this year. If you saw 20 movies during the year, in how many ways can the three best be chosen and ranked?

Key Concept:Permutations of Duplicate Items

• The number of permutations of n items, where p items are identical, q items are identical, r items are identical, and so on, is given by:

• In how many distinct ways can the letters of the word SCIENCE be arranged?

• Find the number of different signals consisting of 9 flags that can be made using 3 white flags, 5 red flags, and 1 blue flag.

• Classwork:

• TB pg. 705/1 – 16 all (omit 11 and 12)

• Remember you must write problems and show ALL work to receive credit for this assignment.

### Combinations

• Combination: a group of items taken without regard to their order.

• Items must be selected from the same group.

• No item is used more than once.

• The order of items makes no difference.

• Distinguish between Permutations and Combinations.

• Six students are running for student government president, vice-president, and treasurer. The student with the greatest number of votes becomes the president, the second highest vice-president, etc. How many different outcomes are possible?

• Permutation/Combination

• Six people are on the board of supervisors for your neighborhood park. A three-person committee is needed to study the possibility of expanding the park. How many different committees could be formed from the six people?

• Permutation/Combination

• Fifty people purchase raffle tickets. Three winning tickets are selected at random. If first prize is \$1,000, second prize is \$500, and third prize is \$100, in how many ways can the prizes be awarded?

Key Concept:Formula for Computing C(n, r)

• Combinations of “n” things taken “r” at a time; the number of possible combinations if “r” items are taken from “n” items is:

• Evaluate the Combination problem.

• Evaluate the Combination problem.

• Evaluate the Combination problem.

• Evaluate the Combination problem.

• Evaluate

• Evaluate

• A mathematics exam consists of 10 multiple-choice questions and 5 open-ended problems in which all work must be shown. If an examinee must answer 8 of the multiple-choice questions and 3 of the open-ended problems, in how many ways can the questions and problems be chosen?

• A book club offers a choice of 8 books from a list of 40. In how many ways can a member make a selection?

• Nine comedy acts will perform over two evenings. Five of the acts will perform on the first evening. How many ways can the schedule for the first evening be made?

• TB pg. 706/35

• TB pg. 707/45

• Classwork:

• TB pg. 18, 20, 24 – 46 Even

• Remember you must write problems and show ALL work to receive credit for the assignment.