1. Permutations Not All Different
Case 3: Ordered Sets of n Objects,

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

3. 2 objects
Permutations Not All Different
Case 3: Ordered Sets of n Objects,
(i.e. some of the objects are the same)

4. 2 objects
Permutations Not All Different
Case 3: Ordered Sets of n Objects,
all different (i.e. some of the objects are the same)
A B
B A

5. 2 objects
Permutations Not All Different
Case 3: Ordered Sets of n Objects,
all different (i.e. some of the objects are the same)
A B
B A
2! 2

6. 2 objects
Permutations Not All Different
Case 3: Ordered Sets of n Objects,
all different 2 same (i.e. some of the objects are the same)
A B A A
B A
2! 2 1

7. 2 objects
Permutations Not All Different
Case 3: Ordered Sets of n Objects,
all different 2 same (i.e. some of the objects are the same)
A B A A
B A
2! 2 1
3 objects

8. 2 objects
Permutations Not All Different
Case 3: Ordered Sets of n Objects,
all different 2 same (i.e. some of the objects are the same)
A B A A
B A
2! 2 1
3 objects
all different
A B C
A C B
B A C
B C A
C A B
C B A

9. 2 objects
Permutations Not All Different
Case 3: Ordered Sets of n Objects,
all different 2 same (i.e. some of the objects are the same)
A B A A
B A
2! 2 1
3 objects
all different
A B C
A C B
B A C
B C A
C A B
C B A
3! 6

10. 2 objects
Permutations Not All Different
Case 3: Ordered Sets of n Objects,
all different 2 same (i.e. some of the objects are the same)
A B A A
B A
2! 2 1
3 objects
all different 2 same
A B C A A B
A C B A B A
B A C B A A
B C A
C A B
C B A
3! 6 3

11. 2 objects
Permutations Not All Different
Case 3: Ordered Sets of n Objects,
all different 2 same (i.e. some of the objects are the same)
A B A A
B A
2! 2 1
3 objects
all different 2 same 3 same
A B C A A B A A A
A C B A B A
B A C B A A
B C A
C A B
C B A
3! 6 3 1

12. 4 objects

13. 4 objects
all different
A B C D C B A D
A B D C C B D A
A C B D C A B D
A C D B C A D B
A D B C C D B A
A D C B C D A B
B A C D D A C B
B A D C D A B C
B C A D D C A B
B C D A D C B A
B D A C D B A C
B D A B D B A B

14. 4 objects
all different
A B C D C B A D
A B D C C B D A
A C B D C A B D
A C D B C A D B
A D B C C D B A
A D C B C D A B
B A C D D A C B
B A D C D A B C
B C A D D C A B
B C D A D C B A
B D A C D B A C
B D A B D B A B
4! 24

15. 4 objects
all different 2 same
A B C D C B A D A A B C
A B D C C B D A A A C B
A C B D C A B D A B A C
A C D B C A D B A B C A
A D B C C D B A A C A B
A D C B C D A B A C B A
B A C D D A C B B A A C
B A D C D A B C B A C A
B C A D D C A B B C A A
B C D A D C B A C A A B
B D A C D B A C C A B A
B D A B D B A B C B A A
4! 24 12

16. 4 objects
all different 2 same 3 same
A B C D C B A D A A B C A A A B
A B D C C B D A A A C B
A A B A
A C B D C A B D A B A C
A B A A
A C D B C A D B A B C A
A D B C C D B A A C A B B A A A
A D C B C D A B A C B A 4
B A C D D A C B B A A C
B A D C D A B C B A C A
B C A D D C A B B C A A
B C D A D C B A C A A B
B D A C D B A C C A B A
B D A B D B A B C B A A
4! 24 12

17. 4 objects
all different 2 same 3 same
A B C D C B A D A A B C A A A B
A B D C C B D A A A C B
A A B A
A C B D C A B D A B A C
A B A A
A C D B C A D B A B C A
A D B C C D B A A C A B B A A A
A D C B C D A B A C B A 4
B A C D D A C B B A A C
B A D C D A B C B A C A
B C A D D C A B B C A A
B C D A D C B A C A A B
B D A C D B A C C A B A 4 same
B D A B D B A B C B A A A A A A
4! 24 12 1

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

19. If we arrange n objects in a line, of which x are alike, the number of
ways we could arrange them are;
n!
Number of Arrangements 
x!

20. If we arrange n objects in a line, of which x are alike, the number of
ways we could arrange them are;
ways of arranging
n! n objects
Number of Arrangements 
x!

21. If we arrange n objects in a line, of which x are alike, the number of
ways we could arrange them are;
ways of arranging
n! n objects
Number of Arrangements 
x! ways of arranging
the like objects

22. If we arrange n objects in a line, of which x are alike, the number of
ways we could arrange them are;
ways of arranging
n! n objects
Number of Arrangements 
x! ways of arranging
the like objects
e.g. How many different words can be formed using all of the
letters in the word
CONNAUGHTON ?

23. If we arrange n objects in a line, of which x are alike, the number of
ways we could arrange them are;
ways of arranging
n! n objects
Number of Arrangements 
x! ways of arranging
the like objects
e.g. How many different words can be formed using all of the
letters in the word
CONNAUGHTON ?
11!
Words 
2!3!

24. If we arrange n objects in a line, of which x are alike, the number of
ways we could arrange them are;
ways of arranging
n! n objects
Number of Arrangements 
x! ways of arranging
the like objects
e.g. How many different words can be formed using all of the
letters in the word
CONNAUGHTON ?
11!
Words 
2!3!
2! for the two O' s

25. If we arrange n objects in a line, of which x are alike, the number of
ways we could arrange them are;
ways of arranging
n! n objects
Number of Arrangements 
x! ways of arranging
the like objects
e.g. How many different words can be formed using all of the
letters in the word
CONNAUGHTON ?
11!
Words 
2!3!
2! for the two O' s 3! for the three N' s

26. If we arrange n objects in a line, of which x are alike, the number of
ways we could arrange them are;
ways of arranging
n! n objects
Number of Arrangements 
x! ways of arranging
the like objects
e.g. How many different words can be formed using all of the
letters in the word
CONNAUGHTON ?
11!
Words 
2!3!
 3326400
2! for the two O' s 3! for the three N' s

27. 2001 Extension 1 HSC Q2c)
The letters A, E, I, O and U are vowels
(i) How many arrangements of the letters in the word ALGEBRAIC
are possible?

28. 2001 Extension 1 HSC Q2c)
The letters A, E, I, O and U are vowels
(i) How many arrangements of the letters in the word ALGEBRAIC
are possible?
9!
Words 
2!

29. 2001 Extension 1 HSC Q2c)
The letters A, E, I, O and U are vowels
(i) How many arrangements of the letters in the word ALGEBRAIC
are possible?
9!
Words 
2!
 181440

30. 2001 Extension 1 HSC Q2c)
The letters A, E, I, O and U are vowels
(i) How many arrangements of the letters in the word ALGEBRAIC
are possible?
9!
Words 
2!
 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?

31. 2001 Extension 1 HSC Q2c)
The letters A, E, I, O and U are vowels
(i) How many arrangements of the letters in the word ALGEBRAIC
are possible?
9!
Words 
2!
 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? 4!
Words   5!
2!

32. 2001 Extension 1 HSC Q2c)
The letters A, E, I, O and U are vowels
(i) How many arrangements of the letters in the word ALGEBRAIC
are possible?
9!
Words 
2!
 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? 4!
Words   5!
2!
Number of ways of
placing the vowels

33. 2001 Extension 1 HSC Q2c)
The letters A, E, I, O and U are vowels
(i) How many arrangements of the letters in the word ALGEBRAIC
are possible?
9!
Words 
2!
 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? 4!
Words   5!
2!
Number of ways of Number of ways of
placing the vowels placing the consonants

34. 2001 Extension 1 HSC Q2c)
The letters A, E, I, O and U are vowels
(i) How many arrangements of the letters in the word ALGEBRAIC
are possible?
9!
Words 
2!
 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? 4!
Words   5!
2!
Number of ways of  1440
Number of ways of
placing the vowels placing the consonants

35. 2001 Extension 1 HSC Q2c)
The letters A, E, I, O and U are vowels
(i) How many arrangements of the letters in the word ALGEBRAIC
are possible?
9!
Words 
2!
 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? 4!
Words   5!
2!
Number of ways of  1440
Number of ways of
placing the vowels placing the consonants
Exercise 10F; odd