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Counting Principle

Counting Principle. 1. A car dealer offers a choice of 6 vinyl top colors, 18 body colors, and 7 upholstery colors. How many color combinations are possible?. 6 * 18 * 7 = 756. Counting Principle.

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Counting Principle

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  1. Counting Principle 1. A car dealer offers a choice of 6 vinyl top colors, 18 body colors, and 7 upholstery colors. How many color combinations are possible? 6 * 18 * 7 = 756

  2. Counting Principle 2. How many ways can six prizes be awarded to 6 children (assuming each child only gets one prize)? What are the possibilities for the first prize? Any of the 6 What are the possibilities for the second prize? Any of the remaining 5 What are the possibilities for the third prize? Any of the 4

  3. Counting Principle 2. How many ways can six prizes be awarded to 6 children (assuming each child only gets one prize)? 6*5*4*3*2*1 = 6! 6! = 720

  4. Counting Principle 2. How many ways can six prizes be awarded to 6 children (assuming each prize can be awarded to any of the 6 children)? What are the possibilities for the first prize? Any of the 6 What are the possibilities for the second prize? Any of the 6 What are the possibilities for the third prize? Any of the 6

  5. Counting Principle 2. How many ways can six prizes be awarded to 6 children (assuming each prize can be awarded to any of the 6 children)? 6*6*6*6*6*6 = 66 66 = 46656

  6. Counting Principle 3. The letters a, b, c, d, and e are used to form 5-letter patterns. How many patterns can be formed if repetitions are allowed? What are the possibilities for the first letter? Any of the 5 What are the possibilities for the second letter? Any of the 5 What are the possibilities for the third letter? Any of the 5

  7. Counting Principle 3. The letters a, b, c, d, and e are used to form 5-letter patterns. How many patterns can be formed if repetitions are allowed? 5*5*5*5*5 = 55 55 = 3125

  8. Counting Principle 3. The letters a, b, c, d, and e are used to form 5-letter patterns. How many patterns can be formed if repetitions are NOT allowed? What are the possibilities for the first letter? Any of the 5 What are the possibilities for the second letter? Any of the remaining 4 What are the possibilities for the third letter? Any of the remaining 3

  9. Counting Principle 3. The letters a, b, c, d, and e are used to form 5-letter patterns. How many patterns can be formed if repetitions are NOT allowed? 5*4*3*2*1 = 5! 5! = 120

  10. Factorial Notation product of all the integers from 1 to n We write "n factorial" with an exclamation mark. Symbol: n! Written as: n! = n(n-1)(n-2)… (3)(2)(1) *(Multiply n times every number between 1 and n)

  11. Try some on your own: a. 9! b. 1! c. _8!_ 6! 4. _19!_ 13! =(9)(8)(7)(6)(5)(4)(3)(2)(1) = 362,880 =(1) = 1 = (8)(7)(6)(5)(4)(3)(2)(1) (6)(5)(4)(3)(2)(1) = 56 = (19)(18)(17)(16)(15)(14)13! 13! = 19,535,040

  12. Permutation An arrangement of n objects in a specific order

  13. Permutation Rule Taking n objects in a specific order using r objects at a time Symbol: nPr Formula: ___n!___ (n-r)! nPr =

  14. Permutation Rule ___n!___ (n-r)! Using : nPr = Find the following: 1. 12P5 2. 20P4 ___12!___ (12-5)! ___20!___ (20-4)! 95040 116280

  15. Permutation Rule Example 1 There are 8 stories for the local news tonight. One will be used as the lead story, one will be the second story, and one will be the closing story. If the director has a total of 8 stories to choose from, how many possible ways can the program be set up? Is there any significance to the order of the 3 stories selected? Lead, 2nd, closing YES! This is why it is a PERMUTATION n= 8 possible stories r = choosing 3 stories (lead, 2nd, closing)

  16. Permutation Rule Example 1 There are 8 stories for the local news tonight. One will be used as the lead story, one will be the second story, and one will be the closing story. If the director has a total of 8 stories to choose from, how many possible ways can the program be set up? _8*7*6*5!_ 5! ___8!___ (8-3)! 336 nPr = 8P3 = = =

  17. Permutation Rule Example 2 How many ways can a chairperson and an assistant chairperson be selected for a research project if there are seven scientists available? Is there any significance to the order of the 2 scientists selected? Chairperson, Assistant Chairperson YES! This is why it is a PERMUTATION n= 7 possible scientists r = choosing 2 scientists (chair, assist. chair)

  18. Permutation Rule Example 2 How many ways can a chairperson and an assistant chairperson be selected for a research project if there are seven scientists available? ___7!___ (7-2)! _7*6*5!_ 5! nPr = 7P2 = 42 = =

  19. Combination A selection of distinct objects without regard to order

  20. Combination Rule The number of combinations of r objects selected from n objects Symbol: nCr Formula: ___n!___ (n-r)!r! nCr =

  21. Combination Rule ___n!___ (n-r)!r! Using : nCr = Find the following: 1. 13C8 2. 9C7 ___13!___ (13-8)!8! ___9!___ (9-7)!7! 1287 36

  22. Combination Rule Example 1 A bicycle shop owner has 12 mountain bicycles in the showroom. The owner wishes to select 5 of them to display at a bicycle show. How many different ways can a group of 5 be selected? Is there any significance to the order of the 5 bicycles selected? NO! This is why it is a COMBINATION n= 12 possible bicycles r = choosing 5 bicycles

  23. Combination Rule Example 1 A bicycle shop owner has 12 mountain bicycles in the showroom. The owner wishes to select 5 of them to display at a bicycle show. How many different ways can a group of 5 be selected? ___12!___ (12-5)!5! _12!_ (7!)5! nCr = 12C5 = = _12*11*10*9*8*7!_ (7!)5! _12*11*10*9*8_ 5*4*3*2*1 = =

  24. Combination Rule Example 1 3 2 _12*11*10*9*8_ 5*4*3*2*1 _12*11*9*8_ 4*3*1 = 5*2 = 10 _12*11*3*2_ 1 792 = =

  25. Combination Rule Example 2 All of the 15 students in a class draw pictures. Only 9 can be displayed at a time. How many different combinations of pictures are there? Is there any significance to the order of the 9 pictures selected? NO! This is why it is a COMBINATION n= 15 possible pictures r = choosing 9 pictures

  26. Combination Rule Example 2 All of the 15 students in a class draw pictures. Only 9 can be displayed at a time. How many different combinations of pictures are there? ___15!___ (15-9)!9! _15!_ (6!)9! nCr = 15C9 = = _15*14*13*12*11*10*9!_ (6!)9! =

  27. Combination Rule Example 2 _15*14*13*12*11*10_ 6*5*4*3*2*1 = _14*13*11*10_ 4*1 = 5005 =

  28. Combination or Permutation? A teacher has 5 different roles to be assigned to the students each day. If there are 18 students in the class, how many possible ways can the 5 roles be assigned? Ask yourself, does order matter or are you just choosing 5 students regardless of order? Order matters, therefore it is a PERMUTATION

  29. Combination or Permutation? A teacher has 5 different roles to be assigned to the students each day. If there are 18 students in the class, how many possible ways can the 5 roles be assigned? Since it is a permutation we use nPr nPr = 18P5 __n!__ (n-r)! __18!__ (18-5)! _18!_ 13! 1,028,160 18P5 = = = =

  30. Combination or Permutation? In a club with 12 members, they need 6 members to represent the club at the schools open house. How many ways can they come up with the 6 students? Ask yourself, does order matter or are you just choosing 6 members regardless of order? Order does not matter, therefore it is a COMBINATION

  31. Combination or Permutation? In a club with 12 members, they need 6 members to represent the club at the schools open house. How many ways can they come up with the 6 students? Since it is a combination we use nCr nCr = 12C6 __n!__ (n-r)!r! __12!___ (12-6)!6! _12!_ 6!6! 924 12C6 = = = =

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