section 12.3 permutations and combinations. permutations

Section 12.3Permutations 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:

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).

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.

!9

45

Example 9: Evaluate the factorial expression.

)!58(

!8

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:

)!(

!Pr

rn

nn

Example 10: 37P

Example 11: 410P

Example 12: 99P

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:

!...!!

!

rqp

n

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.

Combinations

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:

!)!(

!

rrn

nnCr

Example 20: Evaluate the Combination problem.

78C

Example 21: Evaluate the Combination problem.

17C

Example 22: Evaluate the Combination problem.

06C

Example 23: Evaluate the Combination problem.

46

310

C

C

Example 24: Evaluate

220220

!2C

P

Example 25: Evaluate

!44

!46

46

310 C

C

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.