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Section 0-5 Algebra 2

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Permutations and Combinations

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Section 0-5 Algebra 2

  1. 1. PERMUTATIONS AND COMBINATIONS SECTION 0-5
  2. 2. ESSENTIAL QUESTION ‣ How do you solve problems involving permutations and combinations?
  3. 3. VOCABULARY 1. Permutation: 2. Linear Permutation: 3. Combination:
  4. 4. VOCABULARY 1. Permutation: 2. Linear Permutation: 3. Combination: The number of ways in which a set of things can be arranged when order matters
  5. 5. VOCABULARY 1. Permutation: 2. Linear Permutation: 3. Combination: The number of ways in which a set of things can be arranged when order matters The arrangement of objects in a line
  6. 6. VOCABULARY 1. Permutation: 2. Linear Permutation: 3. Combination: The number of ways in which a set of things can be arranged when order matters The arrangement of objects in a line The number of ways in which a set of things can be arranged when order does not matter
  7. 7. FORMULAS 1. Permutation: 2. Permutation with Repetition (n objects of which p and q, etc. are alike: 3. Combination:
  8. 8. FORMULAS 1. Permutation: 2. Permutation with Repetition (n objects of which p and q, etc. are alike: 3. Combination: n Pr = n! (n − r)!
  9. 9. FORMULAS 1. Permutation: 2. Permutation with Repetition (n objects of which p and q, etc. are alike: 3. Combination: n Pr = n! (n − r)! n! p!q!
  10. 10. FORMULAS 1. Permutation: 2. Permutation with Repetition (n objects of which p and q, etc. are alike: 3. Combination: n Pr = n! (n − r)! n! p!q! n Cr = n! (n − r)!r!
  11. 11. EXAMPLE 1 Determine whether each situation involves a permutation or a combination. a. Choosing a 4-digit password b. Selecting 3 movies out of 20 possibilities on Netflix c. Choosing 5 students out of a class of 28 to plan a class party d. Scheduling 6 students to each work an hour selling raffle tickets
  12. 12. EXAMPLE 1 Determine whether each situation involves a permutation or a combination. a. Choosing a 4-digit password b. Selecting 3 movies out of 20 possibilities on Netflix c. Choosing 5 students out of a class of 28 to plan a class party d. Scheduling 6 students to each work an hour selling raffle tickets Permutation
  13. 13. EXAMPLE 1 Determine whether each situation involves a permutation or a combination. a. Choosing a 4-digit password b. Selecting 3 movies out of 20 possibilities on Netflix c. Choosing 5 students out of a class of 28 to plan a class party d. Scheduling 6 students to each work an hour selling raffle tickets Permutation Combination
  14. 14. EXAMPLE 1 Determine whether each situation involves a permutation or a combination. a. Choosing a 4-digit password b. Selecting 3 movies out of 20 possibilities on Netflix c. Choosing 5 students out of a class of 28 to plan a class party d. Scheduling 6 students to each work an hour selling raffle tickets Permutation Combination Combination
  15. 15. EXAMPLE 1 Determine whether each situation involves a permutation or a combination. a. Choosing a 4-digit password b. Selecting 3 movies out of 20 possibilities on Netflix c. Choosing 5 students out of a class of 28 to plan a class party d. Scheduling 6 students to each work an hour selling raffle tickets Permutation Combination Combination Permutation
  16. 16. EXAMPLE 2 Calculate the following permutations and combinations. a. 7 P4
  17. 17. EXAMPLE 2 Calculate the following permutations and combinations. a. 7 P4 = 7! (7− 4)!
  18. 18. EXAMPLE 2 Calculate the following permutations and combinations. a. 7 P4 = 7! (7− 4)! = 7! 3!
  19. 19. EXAMPLE 2 Calculate the following permutations and combinations. a. 7 P4 = 7! (7− 4)! = 7! 3! = 7i 6 i 5i 4 i 3i 2i1 3i 2i1
  20. 20. EXAMPLE 2 Calculate the following permutations and combinations. a. 7 P4 = 7! (7− 4)! = 7! 3! = 7i 6 i 5i 4 i 3i 2i1 3i 2i1
  21. 21. EXAMPLE 2 Calculate the following permutations and combinations. a. 7 P4 = 7! (7− 4)! = 7! 3! = 7i 6 i 5i 4 i 3i 2i1 3i 2i1
  22. 22. EXAMPLE 2 Calculate the following permutations and combinations. a. 7 P4 = 7! (7− 4)! = 7! 3! = 7i 6 i 5i 4 i 3i 2i1 3i 2i1
  23. 23. EXAMPLE 2 Calculate the following permutations and combinations. a. 7 P4 = 7! (7− 4)! = 7! 3! = 7i 6 i 5i 4 i 3i 2i1 3i 2i1 = 840 ways
  24. 24. EXAMPLE 2 Calculate the following permutations and combinations. b. 7 C4
  25. 25. EXAMPLE 2 Calculate the following permutations and combinations. b. 7 C4 = 7! (7− 4)!4!
  26. 26. EXAMPLE 2 Calculate the following permutations and combinations. b. 7 C4 = 7! (7− 4)!4! = 7! 3!4!
  27. 27. EXAMPLE 2 Calculate the following permutations and combinations. b. 7 C4 = 7! (7− 4)!4! = 7! 3!4! = 7i 6 i 5i 4 i 3i 2i1 3i 2i1i 3i 2i1
  28. 28. EXAMPLE 2 Calculate the following permutations and combinations. b. 7 C4 = 7! (7− 4)!4! = 7! 3!4! = 7i 6 i 5i 4 i 3i 2i1 3i 2i1i 3i 2i1
  29. 29. EXAMPLE 2 Calculate the following permutations and combinations. b. 7 C4 = 7! (7− 4)!4! = 7! 3!4! = 7i 6 i 5i 4 i 3i 2i1 3i 2i1i 3i 2i1
  30. 30. EXAMPLE 2 Calculate the following permutations and combinations. b. 7 C4 = 7! (7− 4)!4! = 7! 3!4! = 7i 6 i 5i 4 i 3i 2i1 3i 2i1i 3i 2i1
  31. 31. EXAMPLE 2 Calculate the following permutations and combinations. b. 7 C4 = 7! (7− 4)!4! = 7! 3!4! = 7i 6 i 5i 4 i 3i 2i1 3i 2i1i 3i 2i1
  32. 32. EXAMPLE 2 Calculate the following permutations and combinations. b. 7 C4 = 7! (7− 4)!4! = 7! 3!4! = 7i 6 i 5i 4 i 3i 2i1 3i 2i1i 3i 2i1 = 140 ways
  33. 33. EXAMPLE 3 Fuzzy Jeff has a 4-digit passcode for his iPad. The code is made up of the odd digits: 1, 3, 5, 7, and 9. If each digit can be used only once, how many different codes are available?
  34. 34. EXAMPLE 3 Fuzzy Jeff has a 4-digit passcode for his iPad. The code is made up of the odd digits: 1, 3, 5, 7, and 9. If each digit can be used only once, how many different codes are available? 5 P4
  35. 35. EXAMPLE 3 Fuzzy Jeff has a 4-digit passcode for his iPad. The code is made up of the odd digits: 1, 3, 5, 7, and 9. If each digit can be used only once, how many different codes are available? 5 P4 = 5! (5 − 4)!
  36. 36. EXAMPLE 3 Fuzzy Jeff has a 4-digit passcode for his iPad. The code is made up of the odd digits: 1, 3, 5, 7, and 9. If each digit can be used only once, how many different codes are available? 5 P4 = 5! (5 − 4)! = 5! 1!
  37. 37. EXAMPLE 3 Fuzzy Jeff has a 4-digit passcode for his iPad. The code is made up of the odd digits: 1, 3, 5, 7, and 9. If each digit can be used only once, how many different codes are available? 5 P4 = 5! (5 − 4)! = 5! 1! = 5i 4 i 3i 2
  38. 38. EXAMPLE 3 Fuzzy Jeff has a 4-digit passcode for his iPad. The code is made up of the odd digits: 1, 3, 5, 7, and 9. If each digit can be used only once, how many different codes are available? 5 P4 = 5! (5 − 4)! = 5! 1! = 5i 4 i 3i 2 = 120 codes
  39. 39. EXAMPLE 4 In how many ways can 7 white binders, 5 red binders, and 4 blue binders be arranged on a shelf?
  40. 40. EXAMPLE 4 In how many ways can 7 white binders, 5 red binders, and 4 blue binders be arranged on a shelf? 16! 7!5!4!
  41. 41. EXAMPLE 4 In how many ways can 7 white binders, 5 red binders, and 4 blue binders be arranged on a shelf? 16! 7!5!4! = 16 i15i14 i13i12i11i10 i 9 i 8 5i 4 i 3i 2i1i 4 i 3i 2i1
  42. 42. EXAMPLE 4 In how many ways can 7 white binders, 5 red binders, and 4 blue binders be arranged on a shelf? 16! 7!5!4! = 16 i15i14 i13i12i11i10 i 9 i 8 5i 4 i 3i 2i1i 4 i 3i 2i1
  43. 43. EXAMPLE 4 In how many ways can 7 white binders, 5 red binders, and 4 blue binders be arranged on a shelf? 16! 7!5!4! = 16 i15i14 i13i12i11i10 i 9 i 8 5i 4 i 3i 2i1i 4 i 3i 2i1
  44. 44. EXAMPLE 4 In how many ways can 7 white binders, 5 red binders, and 4 blue binders be arranged on a shelf? 16! 7!5!4! = 16 i15i14 i13i12i11i10 i 9 i 8 5i 4 i 3i 2i1i 4 i 3i 2i1
  45. 45. EXAMPLE 4 In how many ways can 7 white binders, 5 red binders, and 4 blue binders be arranged on a shelf? 16! 7!5!4! = 16 i15i14 i13i12i11i10 i 9 i 8 5i 4 i 3i 2i1i 4 i 3i 2i1 = 14 i13i11i10 i 9 i 8
  46. 46. EXAMPLE 4 In how many ways can 7 white binders, 5 red binders, and 4 blue binders be arranged on a shelf? 16! 7!5!4! = 16 i15i14 i13i12i11i10 i 9 i 8 5i 4 i 3i 2i1i 4 i 3i 2i1 = 14 i13i11i10 i 9 i 8 = 1,441,440 ways
  47. 47. EXAMPLE 5 In how many ways can Shecky choose 3 shirts out of 12 in his closet to pack for a trip?
  48. 48. EXAMPLE 5 In how many ways can Shecky choose 3 shirts out of 12 in his closet to pack for a trip? 12 C3
  49. 49. EXAMPLE 5 In how many ways can Shecky choose 3 shirts out of 12 in his closet to pack for a trip? 12 C3 = 12! (12 − 3)!3!
  50. 50. EXAMPLE 5 In how many ways can Shecky choose 3 shirts out of 12 in his closet to pack for a trip? 12 C3 = 12! (12 − 3)!3! = 12! 9!3!
  51. 51. EXAMPLE 5 In how many ways can Shecky choose 3 shirts out of 12 in his closet to pack for a trip? 12 C3 = 12! (12 − 3)!3! = 12! 9!3! = 12i11i10 3i 2i1
  52. 52. EXAMPLE 5 In how many ways can Shecky choose 3 shirts out of 12 in his closet to pack for a trip? 12 C3 = 12! (12 − 3)!3! = 12! 9!3! = 12i11i10 3i 2i1 = 1320 6
  53. 53. EXAMPLE 5 In how many ways can Shecky choose 3 shirts out of 12 in his closet to pack for a trip? 12 C3 = 12! (12 − 3)!3! = 12! 9!3! = 12i11i10 3i 2i1 = 220 ways = 1320 6
  54. 54. SUMMARY Describe the difference between permutations and combinations.

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