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Permutations - PowerPoint PPT Presentation


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Permutations. Basics. 5!. Does not mean FIVE!. 5! is read five factorial. 5!. = 5 • 4 • 3 • 2 • 1. 7!. = 7 • 6 •5 • 4 • 3 • 2 • 1. 5! = 120. 7! = 5040. When a group of objects or things are arranged in a certain order, the arrangement is called a permutation.

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Presentation Transcript
slide2

Basics

5!

Does not mean FIVE!

5! is read five factorial.

5!

= 5 • 4 • 3 • 2 • 1

7!

= 7 • 6 •5 • 4 • 3 • 2 • 1

5! = 120

7! = 5040

slide3

When a group of objects or

things are arranged in a certain

order, the arrangement is called

a permutation.

The arrangement of objects in a

line is called a linear

permutation.

slide4

Example 1.

A theater owner has 11 films to

show on 8 different screens.

How many different arrange-

ments are there for the 8

screens to show the 11 movies.

slide5

Example 1.

8 screens and 11 movies.

We must make 8 choices.

Choice

1

2

3

4

#

11

10

9

8

Choice

5

6

7

8

#

7

6

5

4

slide6

Example 1.

Choice

1

2

3

4

#

11

10

9

8

Choice

5

6

7

8

#

7

6

5

4

Therefore by the FCP there are

11• 10• 9• 8• 7• 6• 5• 4

slide7

Example 1.

Therefore by the FCP there are

11• 10• 9• 8• 7• 6• 5• 4

arrangements.

The number of way to arrange

11 objects taken 8 at a time is

written as P(11,8).

slide8

n!

(n-r)!

P(n,r) is read as the permutation

of n objects taken r at a time.

Definition of P(n,r)

P(n,r) =

slide9

7!

(7-5)!

7!

2!

Example 2.

Five teens find seven empty

seats at a theater. How many

different seating arrangements

are there?

P(7,5) =

=

slide10

7 • 6 • 5 • 4 • 3 • 2 • 1

2 • 1

7!

(7-5)!

7!

2!

Example 2.

P(7,5) =

=

=

= 7 • 6 • 5 • 4 • 3

= 2520

slide11

Example 3.

A music store manager want to

arrange 5 rock CD’s 4 rap CD’s

and 4 jazz CD’s on a shelf.

How many ways can they be

arranged if they are ordered

according to type?

slide12

Example 3.

The 5 rock CD’s arrange into

P(5,5) or 5! ways.

The 4 rap CD’s arrange into

P(4,4) or 4! ways.

The 4 jazz CD’s arrange into

P(4,4) or 4! ways.

slide13

Example 3.

There are 3 types of CD’s that

arrange into P(3,3) or 3! ways.

The total ways the CD’s can

be arranged is the product.

P(5,5)•P(4,4)•P(4,4)•P(3,3) =

414,720

5!•4!•4!•3! =

slide14

n!

p!q!

Permutations with repetitions.

The number of permutations of

n objects of which p are alike

and q are alike is

The rule can be extended to any

number of objects that repeat.

slide15

13!

2!2!2!

Example 4.

How many ways can the letters

of the word perpendicular be

arranged?

There are 2 p’s, 2 e’s, and 2 r’s.

Therefore

arrangements

slide16

13!

2!2!2!

Example 4.

How many ways can the letters

of the word perpendicular be

arranged?

= 778,377,600

slide17

If n distinct objects are arranged

in a circle there are

or (n-1)! permutations of the

objects around the circle.

n!

n

Circular Permutations.

slide18

Example 5.

A disc jockey is loading a

circular tray with six compact

discs. How many different

ways can these be arranged?

(n-1)! = (6-1)! = 5! = 120

slide19

Circular Permutations (cont).

If n distinct objects are arranged

in a circle and there is a fixed

point on the circle then there are

n! permutations of the objects

around the circle.

slide20

Example 6.

If five different homes are being

built around a cul-de-sac. How

many different arrangements of

the homes are possible?

The cul-de-sac is circular but

the entrance road is a fixed point

slide21

Example 6.

If five different homes are being

built around a cul-de-sac.

The cul-de-sac is circular but

the entrance road is a fixed point

There are n! = 5! =120

arrangements possible.

slide22

We now have several formulas

to remember.

Linear permutations, circular

permutations, circular

permutations with a fixed

point, and permutations with

repetitions.

slide23

One last thing we need to know.

Reflection happens when a

circular permutation can be

flipped over or when a

linear permutation is viewed

from opposite sides. This causes

the number of permutations to

be half as many.