Permutation
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This document provides an overview of permutations including definitions, examples of calculating permutations using factorials and the nPr notation, and applications of permutations such as arranging objects, selecting a subset of objects, and circular arrangements. It discusses permutations of n objects, distinguishable permutations, and permutations of n objects taken r at a time. Examples are provided to illustrate calculating permutations in different scenarios as well as applications and non-applications of permutations.
Permutation
- 1. Teacher III MATHEMATICS 10 Weeks 1-2 of Quarter 3
- 2. Note Please prepare your scientific calculator as it will be used in our online discussion later.
- 3. PERMUTATION An arrangement of objects in a definite order or the ordered arrangement of distinguishable objects without allowing repetitions among the objects.
- 4. Preliminary Task Using your scientific calculator, find the value of the following: 1. 4! 24 2. 3!5! 720 3. 7! 5! 42 4. 8! 2!4! 840 5. 7! 3!4! ∙ 5! 2! 2 100
- 5. What is meant by 𝑛! (𝑛 factorial) 𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2)(𝑛 − 3) … Example: 4! = 4 3 2 1 4! = 24
- 6. Types of Permutation 1. Permutation of 𝑛 objects 2. Distinguishable Permutation 3. Permutation of 𝑛 objects taken 𝑟 at a time 4. Circular Permutation
- 7. Permutation of 𝑛 objects 𝒏𝑷𝒏 = 𝒏! (The permutation of 𝑛 objects is equal to 𝑛 factorial)
- 8. How many arrangements are there? 1. Arranging different 3 portraits on a wall 𝟑! = 𝟔
- 9. How many arrangements are there? 2. Arranging 4 persons in a row for a picture taking 𝟒! = 𝟐𝟒 3. Arranging 5 different figurines in a shelf 𝟓! = 𝟏𝟐𝟎 4. Arranging 6 different potted plants in a row 𝟔! = 𝟕𝟐𝟎 5. Arranging the digits of the number 123456789 𝟗! = 𝟑𝟔𝟐 𝟖𝟖𝟎 6. Arranging the letters of the word CHAIRWOMEN 𝟏𝟎! = 𝟑 𝟔𝟐𝟖 𝟖𝟎𝟎
- 10. Distinguishable Permutation 𝑷 = 𝒏! 𝒑!𝒒!𝒓! There are repeated (or identical) objects in the set.
- 11. How many arrangements are there? 1. Arranging the digits in the number 09778210229 𝑃 = 11! 2! 2! 2! 3! = 831 600 2. Drawing one by one and arranging in a row 4 identical blue, 5 identical yellow, and 3 identical red balls in a bag 𝑃 = 12! 3! 4! 5! = 27 720
- 12. How many arrangements are there? 3. Arranging the letters in the word LOLLIPOP 𝑃 = 8! 3! 2! 2! = 1680 4. Arranging these canned goods 𝑃 = 10! 3! 4! = 25 200
- 13. Permutation of 𝑛 objects taken 𝑟 𝒏𝑷𝒓 = 𝒏! 𝒏−𝒓 ! Permutation of 𝑛 taken 𝑟 at a time where 𝑛 ≥ 𝑟
- 14. How many arrangements are there? 1. Choosing 3 posters to hang on a wall from 5 posters you are keeping 𝒏𝑷𝒓 = 𝒏! 𝒏−𝒓 ! 𝟓𝑷𝟑 = 𝟓! 𝟓−𝟑 ! 𝟓𝑷𝟑 = 𝟓! 𝟐! 𝟓𝑷𝟑 = 𝟔𝟎 Notation: 5P3 / P(5,3) Calculator: 5 shift 𝒏𝑷𝒓 3 =
- 15. How many arrangements are there? 2. Taking two-letter word, without repetition of letters, from the letters of the word COVID Examples: CO, OC, VD, … 𝒏𝑷𝒓 = 𝒏! 𝒏−𝒓 ! 𝟓𝑷𝟐 = 𝟓! 𝟓−𝟐 ! 𝟓𝑷𝟐 = 𝟓! 𝟑! 𝟓𝑷𝟐 = 𝟐𝟎 Notation: 5P2 / P(5,2) Calculator: 5 shift 𝒏𝑷𝒓 2 =
- 16. How many arrangements are there? 3. Taking four-digit numbers, without repetition of digits, from the number 345678 Examples: 3456, 6534, 6745, … 𝒏𝑷𝒓 = 𝒏! 𝒏−𝒓 ! 𝟔𝑷𝟒 = 𝟔! 𝟔−𝟒 ! 𝟔𝑷𝟒 = 𝟔! 𝟐! 𝟔𝑷𝟒 = 𝟑𝟔𝟎 Notation: 6P4 / P(6,4) Calculator: 6 shift 𝒏𝑷𝒓 4 =
- 17. How many arrangements are there? 4. Pirena, Amihan, Alena, and Danaya competing for 1st, 2nd, and 3rd places in spoken poetry Example: Danaya–1st, Alena–2nd, Amihan–3rd 𝒏𝑷𝒓 = 𝒏! 𝒏−𝒓 ! 𝟒𝑷𝟑 = 𝟒! 𝟒−𝟑 ! 𝟒𝑷𝟑 = 𝟒! 𝟏! 𝟒𝑷𝟑 = 𝟐𝟒 Notation: 4P3 / P(4,3) Calculator: 4 shift 𝒏𝑷𝒓 3 =
- 18. How many arrangements are there? 5. Electing Chairperson, Vice Chairperson, Secretary, Treasurer, Auditor, PRO, and Peace Officer from a group of 20 people 𝒏𝑷𝒓 = 𝒏! 𝒏−𝒓 ! 𝟐𝟎𝑷𝟕 = 𝟐𝟎! 𝟐𝟎−𝟕 ! 𝟐𝟎𝑷𝟕 = 𝟐𝟎! 𝟏𝟑! 𝟐𝟎𝑷𝟕 = 𝟑𝟗𝟎 𝟕𝟎𝟎 𝟖𝟎𝟎 Notation: 20P7 / P(20,7) Calculator: 20 shift 𝒏𝑷𝒓 7 =
- 19. Circular Permutation 𝑷 = (𝒏 − 𝟏)!
- 20. How many arrangements are there? How many ways can 5 people sit around a circular table? 𝑃 = (5 − 1)! 𝑃 = 4! 𝑃 = 24
- 22. Other Problems Involving Permutations: 1. There are 3 different History books, 4 different English books, and 8 different Science books. In how many ways can the books be arranged if books of the same subjects must be placed together? 𝑃 = 3! 4! 8! 3! 𝑃 = 34 836 480
- 23. Other Problems Involving Permutations: 2. Three couples want to have their pictures taken. In how many ways can they arrange themselves in a row if couples must stay together? 𝑃 = 3! 2! 𝑃 = 12
- 24. Other Problems Involving Permutations: 3. In how many ways can 8 people arrange themselves in a row if 3 of them insist to stay together? 𝑃 = 6! 3! 𝑃 = 4 320
- 25. Other Problems Involving Permutations: 4. In how many ways can the letters of the word ALGORITHM be arranged if the vowel letters are placed together? Example: AOILGRTHM 𝑃 = 7! 3! 𝑃 = 30 240
- 26. Other Problems Involving Permutations: 5. In how many ways can the letters of the word TIKTOKERIST be arranged if the consonant letters are placed together? Example: TKTKRSTIOEI 𝑃 = 5! 2! ∙ 7! 2! 3! 𝑃 = 25 200
- 27. Other Problems Involving Permutations: 6. In how many ways can 7 people be seated around a circular table if 3 of them insist on sitting beside each other? 𝑃 = 5 − 1 ! ∙ 3! 𝑃 = 144
- 29. Applications of Permutations 1. Using passwords 022988, 228890 2. Using PIN of ATM cards 1357, 3715 3. Winning in a contest Erika – 1st, Carrie – 2nd, Agatha – 3rd Agatha – 1st, Erika – 2nd, Carrie – 3rd
- 30. Applications of Permutations 4. Electing officers in an organization Gerald – President, Julia – VP, Bea – Secretary Bea – President, Gerald – VP, Julia – Secretary 5. Assigning of telephone/mobile numbers 09778210229, 09228210779 6. Assigning plate numbers of vehicles ABY 8512, BAY 1258
- 32. NOT Applications of Permutations 1. Selecting numbers in a lottery 6-12-25-32-34-41, 34-25-41-6-32-12 2. Selecting fruits for salad apple, pineapple, grapes, papaya, pears papaya, pears, apple, pineapple, grapes 3. Choosing members of a committee Marissa, Ellice, Gabriel, Lucinda Lucinda, Gabriel, Ellice, Marissa
- 33. NOT Applications of Permutations 4. Using points on a plane to form a polygon (no three points are collinear) . . . . . . . B A C D E G F
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