12X1 T09 04 permutations II (2010)

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12X1 T09 04 permutations II (2010)

  1. 1. Permutations Not All Different Case 3: Ordered Sets of n Objects,
  2. 2. Permutations Not All Different Case 3: Ordered Sets of n Objects, (i.e. some of the objects are the same)
  3. 3. 2 objects Permutations Not All Different Case 3: Ordered Sets of n Objects, (i.e. some of the objects are the same)
  4. 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. 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. 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. 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. 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. 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. 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. 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. 12. 4 objects
  13. 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. 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. 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. 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 C D B C A D B A B A A 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. 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 C D B C A D B A B A A 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. 18. If we arrange n objects in a line, of which x are alike, the number of ways we could arrange them are;
  19. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

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