Case 3: Ordered Sets of n Objects, Not All DifferentPermutations

Case 3: Ordered Sets of n Objects, Not All Different(i.e. some of the objects are the same)

Permutations

Case 3: Ordered Sets of n Objects, Not All Different(i.e. some of the objects are the same)

2 objectsPermutations

Case 3: Ordered Sets of n Objects, Not All Different(i.e. some of the objects are the same)

2 objectsall different

A B

B A

Permutations

Case 3: Ordered Sets of n Objects, Not All Different(i.e. some of the objects are the same)

2 objectsall different

A B

B A22!

Permutations

Case 3: Ordered Sets of n Objects, Not All Different(i.e. some of the objects are the same)

2 objectsall different

A B

B A22!

2 sameA A

1

Permutations

Case 3: Ordered Sets of n Objects, Not All Different(i.e. some of the objects are the same)

2 objectsall different

A B

B A22!

2 sameA A

13 objects

Permutations

Case 3: Ordered Sets of n Objects, Not All Different(i.e. some of the objects are the same)

2 objectsall different

A B

B A22!

2 sameA A

13 objectsall differentA B C

A C B

B A C

B C A

C A B

C B A

Permutations

Case 3: Ordered Sets of n Objects, Not All Different(i.e. some of the objects are the same)

2 objectsall different

A B

B A22!

2 sameA A

13 objectsall differentA B C

A C B

B A C

B C A

C A B

C B A63!

Permutations

Case 3: Ordered Sets of n Objects, Not All Different(i.e. some of the objects are the same)

2 objectsall different

A B

B A22!

2 sameA A

13 objectsall differentA B C

A C B

B A C

B C A

C A B

C B A63!

2 sameA A B

A B A

B A A

3

Permutations

Case 3: Ordered Sets of n Objects, Not All Different(i.e. some of the objects are the same)

2 objectsall different

A B

B A22!

2 sameA A

13 objectsall differentA B C

A C B

B A C

B C A

C A B

C B A63!

2 sameA A B

A B A

B A A

3

3 sameA A A

1

Permutations

4 objects

4 objectsall different

A B C DA B D CA C B DA C D BA D B CA D C BB A C DB A D CB C A DB C D AB D A CB D A B

C B A DC B D AC A B DC A D BC D B AC D A BD A C BD A B CD C A BD C B AD B A CD B A B

4 objectsall different

A B C DA B D CA C B DA C D BA D B CA D C BB A C DB A D CB C A DB C D AB D A CB D A B

24 4!

C B A DC B D AC A B DC A D BC D B AC D A BD A C BD A B CD C A BD C B AD B A CD B A B

4 objectsall different

A B C DA B D CA C B DA C D BA D B CA D C BB A C DB A D CB C A DB C D AB D A CB D A B

24 4!

2 sameA A B CA A C BA B A CA B C AA C A BA C B AB A A CB A C AB C A AC A A BC A B AC B A A

12

C B A DC B D AC A B DC A D BC D B AC D A BD A C BD A B CD C A BD C B AD B A CD B A B

4 objectsall different

A B C DA B D CA C B DA C D BA D B CA D C BB A C DB A D CB C A DB C D AB D A CB D A B

24 4!

2 sameA A B CA A C BA B A CA B C AA C A BA C B AB A A CB A C AB C A AC A A BC A B AC B A A

12

3 sameA A A B

A A B A

A B A A

B A A A

4

C B A DC B D AC A B DC A D BC D B AC D A BD A C BD A B CD C A BD C B AD B A CD B A B

4 objectsall different

A B C DA B D CA C B DA C D BA D B CA D C BB A C DB A D CB C A DB C D AB D A CB D A B

24 4!

2 sameA A B CA A C BA B A CA B C AA C A BA C B AB A A CB A C AB C A AC A A BC A B AC B A A

12

3 sameA A A B

A A B A

A B A A

B A A A

4

C B A DC B D AC A B DC A D BC D B AC D A BD A C BD A B CD C A BD C B AD B A CD B A B

4 sameA A A A

1

If we arrange n objects in a line, of which x are alike, the number of ways we could arrange them are;

If we arrange n objects in a line, of which x are alike, the number of ways we could arrange them are;

x!n!

tsArrangemen ofNumber

If we arrange n objects in a line, of which x are alike, the number of ways we could arrange them are;

x!n!

tsArrangemen ofNumber objects

arrangingofwaysn

If we arrange n objects in a line, of which x are alike, the number of ways we could arrange them are;

x!n!

tsArrangemen ofNumber objects

arrangingofwaysn

objects like thearrangingofways

If we arrange n objects in a line, of which x are alike, the number of ways we could arrange them are;

x!n!

tsArrangemen ofNumber objects

arrangingofwaysn

objects like thearrangingofways

e.g. How many different words can be formed using all of the letters in the word

CONNAUGHTON ?

If we arrange n objects in a line, of which x are alike, the number of ways we could arrange them are;

x!n!

tsArrangemen ofNumber objects

arrangingofwaysn

objects like thearrangingofways

e.g. How many different words can be formed using all of the letters in the word

CONNAUGHTON ?

!3!2!11 Words

If we arrange n objects in a line, of which x are alike, the number of ways we could arrange them are;

x!n!

tsArrangemen ofNumber objects

arrangingofwaysn

objects like thearrangingofways

e.g. How many different words can be formed using all of the letters in the word

CONNAUGHTON ?

!3!2!11 Words

sO' twofor the2!

If we arrange n objects in a line, of which x are alike, the number of ways we could arrange them are;

x!n!

tsArrangemen ofNumber objects

arrangingofwaysn

objects like thearrangingofways

e.g. How many different words can be formed using all of the letters in the word

CONNAUGHTON ?

!3!2!11 Words

sO' twofor the2! sN'threefor the3!

If we arrange n objects in a line, of which x are alike, the number of ways we could arrange them are;

x!n!

tsArrangemen ofNumber objects

arrangingofwaysn

objects like thearrangingofways

e.g. How many different words can be formed using all of the letters in the word

CONNAUGHTON ?

!3!2!11 Words

sO' twofor the2! sN'threefor the3!3326400

The letters A, E, I, O and U are vowels2001 Extension 1 HSC Q2c)

(i) How many arrangements of the letters in the word ALGEBRAIC are possible?

The letters A, E, I, O and U are vowels2001 Extension 1 HSC Q2c)

(i) How many arrangements of the letters in the word ALGEBRAIC are possible?

!2!9 Words

The letters A, E, I, O and U are vowels2001 Extension 1 HSC Q2c)

(i) How many arrangements of the letters in the word ALGEBRAIC are possible?

!2!9 Words

181440

The letters A, E, I, O and U are vowels2001 Extension 1 HSC Q2c)

(i) How many arrangements of the letters in the word ALGEBRAIC are possible?

!2!9 Words

181440

(ii) How many arrangements of the letters in the word ALGEBRAIC are possible if the vowels must occupy the 2nd, 3rd, 5th, and 8th

positions?

The letters A, E, I, O and U are vowels2001 Extension 1 HSC Q2c)

(i) How many arrangements of the letters in the word ALGEBRAIC are possible?

!2!9 Words

181440

(ii) How many arrangements of the letters in the word ALGEBRAIC are possible if the vowels must occupy the 2nd, 3rd, 5th, and 8th

positions?!2!4 Words !5

The letters A, E, I, O and U are vowels2001 Extension 1 HSC Q2c)

(i) How many arrangements of the letters in the word ALGEBRAIC are possible?

!2!9 Words

181440

(ii) How many arrangements of the letters in the word ALGEBRAIC are possible if the vowels must occupy the 2nd, 3rd, 5th, and 8th

positions?!2!4 Words

vowels theplacingof waysofNumber

!5

The letters A, E, I, O and U are vowels2001 Extension 1 HSC Q2c)

(i) How many arrangements of the letters in the word ALGEBRAIC are possible?

!2!9 Words

181440

(ii) How many arrangements of the letters in the word ALGEBRAIC are possible if the vowels must occupy the 2nd, 3rd, 5th, and 8th

positions?!2!4 Words

vowels theplacingof waysofNumber

consonants theplacing ofwaysofNumber

!5

The letters A, E, I, O and U are vowels2001 Extension 1 HSC Q2c)

(i) How many arrangements of the letters in the word ALGEBRAIC are possible?

!2!9 Words

181440

(ii) How many arrangements of the letters in the word ALGEBRAIC are possible if the vowels must occupy the 2nd, 3rd, 5th, and 8th

positions?!2!4 Words

vowels theplacingof waysofNumber

consonants theplacing ofwaysofNumber

!5

1440

The letters A, E, I, O and U are vowels2001 Extension 1 HSC Q2c)

(i) How many arrangements of the letters in the word ALGEBRAIC are possible?

!2!9 Words

181440

(ii) How many arrangements of the letters in the word ALGEBRAIC are possible if the vowels must occupy the 2nd, 3rd, 5th, and 8th

positions?!2!4 Words

vowels theplacingof waysofNumber

consonants theplacing ofwaysofNumber

!5

1440

Exercise 10F; odd