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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