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### Section 12.3

Permutations and Combinations

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:
- 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).

- An ordered arrangement for items that occurs when

Key Term:

- 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

Example 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?

Example 2: Counting Permutations

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

Example 3:

- 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?

Example 4:

- Evaluate the factorial expression.
- 12!10!

Example 5:

- Evaluate the factorial expression.
- 29!25!

Example 6:

- Evaluate the factorial expression.
- 6! – 3!

Example 7:

- Evaluate the factorial expression.
- (6 – 3)!

Example 8:

- Evaluate the factorial expression.

Example 9:

- 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:

Example 13:

- 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?

Example 14:

- 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:

Example 15:

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

Example 16:

- 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.

Section 12.3 Assignment

- 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.

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

Key Term:

- 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.

Example 17:

- 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?

Example 18:

- 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?

Example 19:

- 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:

Example 20:

- Evaluate the Combination problem.

Example 21:

- Evaluate the Combination problem.

Example 22:

- Evaluate the Combination problem.

Example 23:

- Evaluate the Combination problem.

Example 24:

- Evaluate

Example 25:

- Evaluate

Example 26:

- 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?

Example 27:

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

Example 28:

- 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?

Example 29:

- TB pg. 706/35

Example 30:

- TB pg. 707/45

Section 12.3 Assignment

- Classwork:
- TB pg. 18, 20, 24 – 46 Even
- Remember you must write problems and show ALL work to receive credit for the assignment.

- TB pg. 18, 20, 24 – 46 Even

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