Case 4: Ordered Sets of n Objects, Arranged in a Circle What is the difference between placing objects in a line and placing

objects in a circle?

Permutations

Case 4: Ordered Sets of n Objects, Arranged in a Circle What is the difference between placing objects in a line and placing

objects in a circle?

The difference is the number of ways the first object can be placed.

Permutations

Case 4: Ordered Sets of n Objects, Arranged in a Circle What is the difference between placing objects in a line and placing

objects in a circle?

The difference is the number of ways the first object can be placed.

Line

Permutations

Case 4: Ordered Sets of n Objects, Arranged in a Circle What is the difference between placing objects in a line and placing

objects in a circle?

The difference is the number of ways the first object can be placed.

Line

Permutations

Case 4: Ordered Sets of n Objects, Arranged in a Circle What is the difference between placing objects in a line and placing

objects in a circle?

The difference is the number of ways the first object can be placed.

Line

In a line there is a definite start and finish of the line.

Permutations

Case 4: Ordered Sets of n Objects, Arranged in a Circle What is the difference between placing objects in a line and placing

objects in a circle?

The difference is the number of ways the first object can be placed.

Line

In a line there is a definite start and finish of the line.

The first object has a choice of 6 positions

Permutations

Case 4: Ordered Sets of n Objects, Arranged in a Circle What is the difference between placing objects in a line and placing

objects in a circle?

The difference is the number of ways the first object can be placed.

Line

In a line there is a definite start and finish of the line.

The first object has a choice of 6 positions

Circle

Permutations

Case 4: Ordered Sets of n Objects, Arranged in a Circle What is the difference between placing objects in a line and placing

objects in a circle?

The difference is the number of ways the first object can be placed.

Line

In a line there is a definite start and finish of the line.

The first object has a choice of 6 positions

Circle

Permutations

Case 4: Ordered Sets of n Objects, Arranged in a Circle What is the difference between placing objects in a line and placing

objects in a circle?

The difference is the number of ways the first object can be placed.

Line

In a line there is a definite start and finish of the line.

The first object has a choice of 6 positions

Circle In a circle there is no definite start or

finish of the circle.

Permutations

Case 4: Ordered Sets of n Objects, Arranged in a Circle What is the difference between placing objects in a line and placing

objects in a circle?

The difference is the number of ways the first object can be placed.

Line

In a line there is a definite start and finish of the line.

The first object has a choice of 6 positions

Circle In a circle there is no definite start or

finish of the circle.

It is not until the first object chooses its

position that positions are defined.

Permutations

Line

Circle

Line

n tsArrangemen ofNumber

1object for

iespossibilit

1 tsArrangemen ofNumber

1object for

iespossibilit

Circle

Line

n tsArrangemen ofNumber

1object for

iespossibilit

1 n

2object for

iespossibilit

Circle

1 tsArrangemen ofNumber

1object for

iespossibilit

1 n

2object for

iespossibilit

Line

n tsArrangemen ofNumber

1object for

iespossibilit

1 n 2 n

2object for

iespossibilit

3object for

iespossibilit

Circle

1 tsArrangemen ofNumber

1object for

iespossibilit

1 n 2 n

2object for

iespossibilit

3object for

iespossibilit

Line

n tsArrangemen ofNumber

1object for

iespossibilit

1 n 2 n 1

2object for

iespossibilit

3object for

iespossibilit

objectlast for

iespossibilit

Circle

1 tsArrangemen ofNumber

1object for

iespossibilit

1 n 2 n 1

2object for

iespossibilit

3object for

iespossibilit

objectlast for

iespossibilit

Line

Circle

n tsArrangemen ofNumber

1object for

iespossibilit

1 n 2 n 1

2object for

iespossibilit

3object for

iespossibilit

objectlast for

iespossibilit

1 tsArrangemen ofNumber

1object for

iespossibilit

1 n 2 n 1

2object for

iespossibilit

3object for

iespossibilit

objectlast for

iespossibilit

n

n! circle ain tsArrangemen ofNumber

Line

Circle

n tsArrangemen ofNumber

1object for

iespossibilit

1 n 2 n 1

2object for

iespossibilit

3object for

iespossibilit

objectlast for

iespossibilit

1 tsArrangemen ofNumber

1object for

iespossibilit

1 n 2 n 1

2object for

iespossibilit

3object for

iespossibilit

objectlast for

iespossibilit

n

n! circle ain tsArrangemen ofNumber

!1 n

e.g. A meeting room contains a round table surrounded by ten

chairs.

(i) A committee of ten people includes three teenagers. How

many arrangements are there in which all three sit

together?

e.g. A meeting room contains a round table surrounded by ten

chairs.

(i) A committee of ten people includes three teenagers. How

many arrangements are there in which all three sit

together?

!3 tsArrangemen

arranged

becan teenagersthree

the waysofnumber the

e.g. A meeting room contains a round table surrounded by ten

chairs.

(i) A committee of ten people includes three teenagers. How

many arrangements are there in which all three sit

together?

!3 tsArrangemen

arranged

becan teenagersthree

the waysofnumber the

!7others 7 ) teenagers(3

circle ain objects 8

arranging of waysofnumber

e.g. A meeting room contains a round table surrounded by ten

chairs.

(i) A committee of ten people includes three teenagers. How

many arrangements are there in which all three sit

together?

!3 tsArrangemen

arranged

becan teenagersthree

the waysofnumber the

!7others 7 ) teenagers(3

circle ain objects 8

arranging of waysofnumber

30240

(ii) Elections are held for Chairperson and Secretary.

What is the probability that they are seated directly opposite each

other?

(ii) Elections are held for Chairperson and Secretary.

What is the probability that they are seated directly opposite each

other?

!9 ns)restrictio (no Ways

(ii) Elections are held for Chairperson and Secretary.

What is the probability that they are seated directly opposite each

other?

!9 ns)restrictio (no Ways

1 ons)(restricti Ways

circle in the

1st are they as

anywheresit

canPresident

(ii) Elections are held for Chairperson and Secretary.

What is the probability that they are seated directly opposite each

other?

!9 ns)restrictio (no Ways

1 ons)(restricti Ways

circle in the

1st are they as

anywheresit

canPresident

1

President

oppositesit

mustSecretary

(ii) Elections are held for Chairperson and Secretary.

What is the probability that they are seated directly opposite each

other?

!9 ns)restrictio (no Ways

1 ons)(restricti Ways

circle in the

1st are they as

anywheresit

canPresident

1 !8

President

oppositesit

mustSecretary

gocan people

remaining Ways

(ii) Elections are held for Chairperson and Secretary.

What is the probability that they are seated directly opposite each

other?

!9 ns)restrictio (no Ways

1 ons)(restricti Ways

circle in the

1st are they as

anywheresit

canPresident

1 !8

President

oppositesit

mustSecretary

gocan people

remaining Ways

!9

!811opposite S & P

P

(ii) Elections are held for Chairperson and Secretary.

What is the probability that they are seated directly opposite each

other?

!9 ns)restrictio (no Ways

1 ons)(restricti Ways

circle in the

1st are they as

anywheresit

canPresident

1 !8

President

oppositesit

mustSecretary

gocan people

remaining Ways

!9

!811opposite S & P

P

9

1

(ii) Elections are held for Chairperson and Secretary.

What is the probability that they are seated directly opposite each

other?

!9 ns)restrictio (no Ways

1 ons)(restricti Ways

circle in the

1st are they as

anywheresit

canPresident

1 !8

President

oppositesit

mustSecretary

gocan people

remaining Ways

!9

!811opposite S & P

P

9

1

Note: of 9 seats only

1 is opposite the

President

9

1 oppositeP

Sometimes simple

logic is quicker!!!!

Seven people are to be seated at a round table

2002 Extension 1 HSC Q3a)

(i) How many seating arrangements are possible?

Seven people are to be seated at a round table

2002 Extension 1 HSC Q3a)

(i) How many seating arrangements are possible?

!6 tsArrangemen

Seven people are to be seated at a round table

2002 Extension 1 HSC Q3a)

(i) How many seating arrangements are possible?

!6 tsArrangemen

720

Seven people are to be seated at a round table

2002 Extension 1 HSC Q3a)

(i) How many seating arrangements are possible?

(ii) Two people, Kevin and Jill, refuse to sit next to each other. How

many seating arrangements are then possible?

!6 tsArrangemen

720

Seven people are to be seated at a round table

2002 Extension 1 HSC Q3a)

(i) How many seating arrangements are possible?

(ii) Two people, Kevin and Jill, refuse to sit next to each other. How

many seating arrangements are then possible?

!6 tsArrangemen

720

Note: it is easier to work out the number of ways Kevin and Jill are

together and subtract from total number of arrangements.

Seven people are to be seated at a round table

2002 Extension 1 HSC Q3a)

(i) How many seating arrangements are possible?

(ii) Two people, Kevin and Jill, refuse to sit next to each other. How

many seating arrangements are then possible?

!6 tsArrangemen

720

Note: it is easier to work out the number of ways Kevin and Jill are

together and subtract from total number of arrangements.

!2 tsArrangemen

togetherare Jill &Kevin

waysofnumber the

Seven people are to be seated at a round table

2002 Extension 1 HSC Q3a)

(i) How many seating arrangements are possible?

(ii) Two people, Kevin and Jill, refuse to sit next to each other. How

many seating arrangements are then possible?

!6 tsArrangemen

720

Note: it is easier to work out the number of ways Kevin and Jill are

together and subtract from total number of arrangements.

!2 tsArrangemen

togetherare Jill &Kevin

waysofnumber the

others 5 Jill) &(Kevin

circle ain objects 6

arranging of waysofnumber

!5

Seven people are to be seated at a round table

2002 Extension 1 HSC Q3a)

(i) How many seating arrangements are possible?

(ii) Two people, Kevin and Jill, refuse to sit next to each other. How

many seating arrangements are then possible?

!6 tsArrangemen

720

Note: it is easier to work out the number of ways Kevin and Jill are

together and subtract from total number of arrangements.

!2 tsArrangemen

togetherare Jill &Kevin

waysofnumber the

others 5 Jill) &(Kevin

circle ain objects 6

arranging of waysofnumber

240

!5

Seven people are to be seated at a round table

2002 Extension 1 HSC Q3a)

(i) How many seating arrangements are possible?

(ii) Two people, Kevin and Jill, refuse to sit next to each other. How

many seating arrangements are then possible?

!6 tsArrangemen

720

Note: it is easier to work out the number of ways Kevin and Jill are

together and subtract from total number of arrangements.

!2 tsArrangemen

togetherare Jill &Kevin

waysofnumber the

others 5 Jill) &(Kevin

circle ain objects 6

arranging of waysofnumber

240

!5240720 tsArrangemen

480

Seven people are to be seated at a round table

2002 Extension 1 HSC Q3a)

(i) How many seating arrangements are possible?

(ii) Two people, Kevin and Jill, refuse to sit next to each other. How

many seating arrangements are then possible?

!6 tsArrangemen

720

Note: it is easier to work out the number of ways Kevin and Jill are

together and subtract from total number of arrangements.

!2 tsArrangemen

togetherare Jill &Kevin

waysofnumber the

others 5 Jill) &(Kevin

circle ain objects 6

arranging of waysofnumber

240

!5240720 tsArrangemen

480

A security lock has 8 buttons labelled as shown. Each person using

the lock is given a 3 letter code.

1995 Extension 1 HSC Q3a)

(i) How many different codes are possible if letters can be repeated

and their order is important?

A B C D E F G H

A security lock has 8 buttons labelled as shown. Each person using

the lock is given a 3 letter code.

1995 Extension 1 HSC Q3a)

(i) How many different codes are possible if letters can be repeated

and their order is important?

With replacement

Use Basic Counting Principle

A B C D E F G H

A security lock has 8 buttons labelled as shown. Each person using

the lock is given a 3 letter code.

1995 Extension 1 HSC Q3a)

(i) How many different codes are possible if letters can be repeated

and their order is important?

888 Codes With replacement

Use Basic Counting Principle

A B C D E F G H

A security lock has 8 buttons labelled as shown. Each person using

the lock is given a 3 letter code.

1995 Extension 1 HSC Q3a)

(i) How many different codes are possible if letters can be repeated

and their order is important?

888 Codes With replacement

Use Basic Counting Principle 512

A B C D E F G H

A security lock has 8 buttons labelled as shown. Each person using

the lock is given a 3 letter code.

1995 Extension 1 HSC Q3a)

(i) How many different codes are possible if letters can be repeated

and their order is important?

888 Codes With replacement

Use Basic Counting Principle 512

A B C D E F G H

(ii) How many different codes are possible if letters cannot be

repeated and their order is important?

A security lock has 8 buttons labelled as shown. Each person using

the lock is given a 3 letter code.

1995 Extension 1 HSC Q3a)

(i) How many different codes are possible if letters can be repeated

and their order is important?

888 Codes With replacement

Use Basic Counting Principle 512

Without replacement

Order is important

Permutation

A B C D E F G H

(ii) How many different codes are possible if letters cannot be

repeated and their order is important?

A security lock has 8 buttons labelled as shown. Each person using

the lock is given a 3 letter code.

1995 Extension 1 HSC Q3a)

(i) How many different codes are possible if letters can be repeated

and their order is important?

888 Codes With replacement

Use Basic Counting Principle 512

3

8 Codes P Without replacement

Order is important

Permutation

A B C D E F G H

(ii) How many different codes are possible if letters cannot be

repeated and their order is important?

A security lock has 8 buttons labelled as shown. Each person using

the lock is given a 3 letter code.

1995 Extension 1 HSC Q3a)

(i) How many different codes are possible if letters can be repeated

and their order is important?

888 Codes With replacement

Use Basic Counting Principle 512

3

8 Codes P

336

Without replacement

Order is important

Permutation

A B C D E F G H

(ii) How many different codes are possible if letters cannot be

repeated and their order is important?

(iii) Now suppose that the lock operates by holding 3 buttons down

together, so that the order is NOT important.

How many different codes are possible?

(iii) Now suppose that the lock operates by holding 3 buttons down

together, so that the order is NOT important.

How many different codes are possible?

Without replacement

Order is not important

Combination

(iii) Now suppose that the lock operates by holding 3 buttons down

together, so that the order is NOT important.

How many different codes are possible?

3

8 Codes C Without replacement

Order is not important

Combination

(iii) Now suppose that the lock operates by holding 3 buttons down

together, so that the order is NOT important.

How many different codes are possible?

3

8 Codes C

56

Without replacement

Order is not important

Combination

(iii) Now suppose that the lock operates by holding 3 buttons down

together, so that the order is NOT important.

How many different codes are possible?

3

8 Codes C

56

Without replacement

Order is not important

Combination

Exercise 10I; odds