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
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
