5.5 permutations and combinations

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5.5 permutations and combinations

  1. 1. Permutations and Combinations
  2. 2. Permutations and CombinationsA k-permutation is an ordered lineup of k objects.
  3. 3. Permutations and CombinationsA k-permutation is an ordered lineup of k objects.Example A. List all the 2-permutations taken from {a, b, c}.
  4. 4. Permutations and CombinationsA k-permutation is an ordered lineup of k objects.Example A. List all the 2-permutations taken from {a, b, c}.ab, ba, ac, ca, bc, cb
  5. 5. Permutations and CombinationsA k-permutation is an ordered lineup of k objects.Example A. List all the 2-permutations taken from {a, b, c}.ab, ba, ac, ca, bc, cbExample B. List all the 3-permutations taken from {a, b, c}.
  6. 6. Permutations and CombinationsA k-permutation is an ordered lineup of k objects.Example A. List all the 2-permutations taken from {a, b, c}.ab, ba, ac, ca, bc, cbExample B. List all the 3-permutations taken from {a, b, c}.abc, acb, bac, bca, cab, cba
  7. 7. Permutations and CombinationsA k-permutation is an ordered lineup of k objects.Example A. List all the 2-permutations taken from {a, b, c}.ab, ba, ac, ca, bc, cbExample B. List all the 3-permutations taken from {a, b, c}.abc, acb, bac, bca, cab, cbaThe number of k-permutations (ordered arrangements) takenfrom n objects is: n!nPk = (n – k)!
  8. 8. Permutations and CombinationsA k-permutation is an ordered lineup of k objects.Example A. List all the 2-permutations taken from {a, b, c}.ab, ba, ac, ca, bc, cbExample B. List all the 3-permutations taken from {a, b, c}.abc, acb, bac, bca, cab, cbaThe number of k-permutations (ordered arrangements) takenfrom n objects is: n!nPk = (n – k)!Example C. How many 2-permutations taken from {a, b, c}are there?
  9. 9. Permutations and CombinationsA k-permutation is an ordered lineup of k objects.Example A. List all the 2-permutations taken from {a, b, c}.ab, ba, ac, ca, bc, cbExample B. List all the 3-permutations taken from {a, b, c}.abc, acb, bac, bca, cab, cbaThe number of k-permutations (ordered arrangements) takenfrom n objects is: n!nPk = (n – k)!Example C. How many 2-permutations taken from {a, b, c}are there?n=3, k=2,
  10. 10. Permutations and CombinationsA k-permutation is an ordered lineup of k objects.Example A. List all the 2-permutations taken from {a, b, c}.ab, ba, ac, ca, bc, cbExample B. List all the 3-permutations taken from {a, b, c}.abc, acb, bac, bca, cab, cbaThe number of k-permutations (ordered arrangements) takenfrom n objects is: n!nPk = (n – k)!Example C. How many 2-permutations taken from {a, b, c}are there? 3!n=3, k=2, 3P2 = (3 – 2)!
  11. 11. Permutations and CombinationsA k-permutation is an ordered lineup of k objects.Example A. List all the 2-permutations taken from {a, b, c}.ab, ba, ac, ca, bc, cbExample B. List all the 3-permutations taken from {a, b, c}.abc, acb, bac, bca, cab, cbaThe number of k-permutations (ordered arrangements) takenfrom n objects is: n!nPk = (n – k)!Example C. How many 2-permutations taken from {a, b, c}are there? 3! 6n=3, k=2, 3P2 = (3 – 2)! = 1 = 6
  12. 12. Permutations and CombinationsA k-permutation is an ordered lineup of k objects.Example A. List all the 2-permutations taken from {a, b, c}.ab, ba, ac, ca, bc, cbExample B. List all the 3-permutations taken from {a, b, c}.abc, acb, bac, bca, cab, cbaThe number of k-permutations (ordered arrangements) takenfrom n objects is: n!nPk = (n – k)!Example C. How many 2-permutations taken from {a, b, c}are there? 3! 6n=3, k=2, 3P2 = (3 – 2)! = 1 = 6They are {ab, ba, ac, ca, bc, cb}.
  13. 13. Permutations and CombinationsExample D. How many different arrangements of 7 peoplefrom a group of 10 people in a row of 7 seats are there?
  14. 14. Permutations and CombinationsExample D. How many different arrangements of 7 peoplefrom a group of 10 people in a row of 7 seats are there?There are 10 people so n = 10.
  15. 15. Permutations and CombinationsExample D. How many different arrangements of 7 peoplefrom a group of 10 people in a row of 7 seats are there?There are 10 people so n = 10. We are to seat 7 of them inorder so k = 7,
  16. 16. Permutations and CombinationsExample D. How many different arrangements of 7 peoplefrom a group of 10 people in a row of 7 seats are there?There are 10 people so n = 10. We are to seat 7 of them inorder so k = 7, so there are10P7 = 10! / (10 – 7)!
  17. 17. Permutations and CombinationsExample D. How many different arrangements of 7 peoplefrom a group of 10 people in a row of 7 seats are there?There are 10 people so n = 10. We are to seat 7 of them inorder so k = 7, so there are10P7 = 10! / (10 – 7)! =10! / 3!
  18. 18. Permutations and CombinationsExample D. How many different arrangements of 7 peoplefrom a group of 10 people in a row of 7 seats are there?There are 10 people so n = 10. We are to seat 7 of them inorder so k = 7, so there are10P7 = 10! / (10 – 7)! =10! / 3! = 10 x 9 x 8 x .. X 4 = 604800 possibilities.
  19. 19. Permutations and CombinationsExample D. How many different arrangements of 7 peoplefrom a group of 10 people in a row of 7 seats are there?There are 10 people so n = 10. We are to seat 7 of them inorder so k = 7, so there are10P7 = 10! / (10 – 7)! =10! / 3! = 10 x 9 x 8 x .. X 4 = 604800 possibilities.A k-combination is a (unordered) collection of k objects.
  20. 20. Permutations and CombinationsExample D. How many different arrangements of 7 peoplefrom a group of 10 people in a row of 7 seats are there?There are 10 people so n = 10. We are to seat 7 of them inorder so k = 7, so there are10P7 = 10! / (10 – 7)! =10! / 3! = 10 x 9 x 8 x .. X 4 = 604800 possibilities.A k-combination is a (unordered) collection of k objects.Example E. List all the 2-combinations from the set {a, b, c}.
  21. 21. Permutations and CombinationsExample D. How many different arrangements of 7 peoplefrom a group of 10 people in a row of 7 seats are there?There are 10 people so n = 10. We are to seat 7 of them inorder so k = 7, so there are10P7 = 10! / (10 – 7)! =10! / 3! = 10 x 9 x 8 x .. X 4 = 604800 possibilities.A k-combination is a (unordered) collection of k objects.Example E. List all the 2-combinations from the set {a, b, c}.{a, b}, {a, c}, {b, c}
  22. 22. Permutations and CombinationsExample D. How many different arrangements of 7 peoplefrom a group of 10 people in a row of 7 seats are there?There are 10 people so n = 10. We are to seat 7 of them inorder so k = 7, so there are10P7 = 10! / (10 – 7)! =10! / 3! = 10 x 9 x 8 x .. X 4 = 604800 possibilities.A k-combination is a (unordered) collection of k objects.Example E. List all the 2-combinations from the set {a, b, c}.{a, b}, {a, c}, {b, c}Example F. List all the 3-combination taken from {a, b, c}.
  23. 23. Permutations and CombinationsExample D. How many different arrangements of 7 peoplefrom a group of 10 people in a row of 7 seats are there?There are 10 people so n = 10. We are to seat 7 of them inorder so k = 7, so there are10P7 = 10! / (10 – 7)! =10! / 3! = 10 x 9 x 8 x .. X 4 = 604800 possibilities.A k-combination is a (unordered) collection of k objects.Example E. List all the 2-combinations from the set {a, b, c}.{a, b}, {a, c}, {b, c}Example F. List all the 3-combination taken from {a, b, c}.{a, b, c}
  24. 24. Permutations and CombinationsExample D. How many different arrangements of 7 peoplefrom a group of 10 people in a row of 7 seats are there?There are 10 people so n = 10. We are to seat 7 of them inorder so k = 7, so there are10P7 = 10! / (10 – 7)! =10! / 3! = 10 x 9 x 8 x .. X 4 = 604800 possibilities.A k-combination is a (unordered) collection of k objects.Example E. List all the 2-combinations from the set {a, b, c}.{a, b}, {a, c}, {b, c}Example F. List all the 3-combination taken from {a, b, c}.{a, b, c}The number of k-combinations (unordered collections) takenfrom n objects is: n!nCk = (n – k)!k!
  25. 25. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?
  26. 26. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2,
  27. 27. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2, hence there are C2 = 3!3 (3 – 2)!2!
  28. 28. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2, hence there are 3! 33C2 = (3 – 2)!2!
  29. 29. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2, hence there are 3! 33C2 = =3 (3 – 2)!2!
  30. 30. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2, hence there are 3! 33C2 = =3 (3 – 2)!2!So there are three 2-combinations.
  31. 31. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2, hence there are 3! 33C2 = =3 (3 – 2)!2!So there are three 2-combinations.They are {a, b}, {a, c}, {b, c}.
  32. 32. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2, hence there are 3! 33C2 = =3 (3 – 2)!2!So there are three 2-combinations.They are {a, b}, {a, c}, {b, c}.Example H. A Chinese take-out offers a beef dish, a chickendish, a vegetable dish, fried rice, and fried noodles. We maychose any 3 of them for a 3-Combo Special. How manydifferent 3-Combo Specials are possible?
  33. 33. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2, hence there are 3! 33C2 = =3 (3 – 2)!2!So there are three 2-combinations.They are {a, b}, {a, c}, {b, c}.Example H. A Chinese take-out offers a beef dish, a chickendish, a vegetable dish, fried rice, and fried noodles. We maychose any 3 of them for a 3-Combo Special. How manydifferent 3-Combo Specials are possible?n = 5, we are to take 3 of so k = 3,
  34. 34. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2, hence there are 3! 33C2 = =3 (3 – 2)!2!So there are three 2-combinations.They are {a, b}, {a, c}, {b, c}.Example H. A Chinese take-out offers a beef dish, a chickendish, a vegetable dish, fried rice, and fried noodles. We maychose any 3 of them for a 3-Combo Special. How manydifferent 3-Combo Specials are possible?n = 5, we are to take 3 of so k = 3, hence there are 5!5 C3 = (5 – 3)!3!
  35. 35. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2, hence there are 3! 33C2 = =3 (3 – 2)!2!So there are three 2-combinations.They are {a, b}, {a, c}, {b, c}.Example H. A Chinese take-out offers a beef dish, a chickendish, a vegetable dish, fried rice, and fried noodles. We maychose any 3 of them for a 3-Combo Special. How manydifferent 3-Combo Specials are possible?n = 5, we are to take 3 of so k = 3, hence there are 5! 5!5 C3 = (5 – 3)!3! = 2!3!
  36. 36. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2, hence there are 3! 33C2 = =3 (3 – 2)!2!So there are three 2-combinations.They are {a, b}, {a, c}, {b, c}.Example H. A Chinese take-out offers a beef dish, a chickendish, a vegetable dish, fried rice, and fried noodles. We maychose any 3 of them for a 3-Combo Special. How manydifferent 3-Combo Specials are possible?n = 5, we are to take 3 of so k = 3, hence there are 5! 5! 5*45C3 = (5 – 3)!3! = 2!3!
  37. 37. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2, hence there are 3! 33C2 = =3 (3 – 2)!2!So there are three 2-combinations.They are {a, b}, {a, c}, {b, c}.Example H. A Chinese take-out offers a beef dish, a chickendish, a vegetable dish, fried rice, and fried noodles. We maychose any 3 of them for a 3-Combo Special. How manydifferent 3-Combo Specials are possible?n = 5, we are to take 3 of so k = 3, hence there are 5! 5! 5*45C3 = (5 – 3)!3! = 2!3! = 10 3-Combos specials.
  38. 38. Permutations and CombinationsExample I. There are 5 men and 8 women.a. We are to select a president, a vice president, a manager,a treasurer from them. How many different possibilities arethere?
  39. 39. Permutations and CombinationsExample I. There are 5 men and 8 women.a. We are to select a president, a vice president, a manager,a treasurer from them. How many different possibilities arethere?These are permutations since the order is important.
  40. 40. Permutations and CombinationsExample I. There are 5 men and 8 women.a. We are to select a president, a vice president, a manager,a treasurer from them. How many different possibilities arethere?These are permutations since the order is important.Hence there are 13P4 possibilities.
  41. 41. Permutations and CombinationsExample I. There are 5 men and 8 women.a. We are to select a president, a vice president, a manager,a treasurer from them. How many different possibilities arethere?These are permutations since the order is important.Hence there are 13P4 possibilities.b. We are to select 4 people for a committee, how manydifferent 4-people committees are possible?
  42. 42. Permutations and CombinationsExample I. There are 5 men and 8 women.a. We are to select a president, a vice president, a manager,a treasurer from them. How many different possibilities arethere?These are permutations since the order is important.Hence there are 13P4 possibilities.b. We are to select 4 people for a committee, how manydifferent 4-people committees are possible?These are combinations so there are 13C4 possibilities.
  43. 43. Permutations and CombinationsExample I. There are 5 men and 8 women.a. We are to select a president, a vice president, a manager,a treasurer from them. How many different possibilities arethere?These are permutations since the order is important.Hence there are 13P4 possibilities.b. We are to select 4 people for a committee, how manydifferent 4-people committees are possible?These are combinations so there are 13C4 possibilities.c.We are to select two men and two women for a 4-peoplecommittee, how many are possibilities?
  44. 44. Permutations and CombinationsExample I. There are 5 men and 8 women.a. We are to select a president, a vice president, a manager,a treasurer from them. How many different possibilities arethere?These are permutations since the order is important.Hence there are 13P4 possibilities.b. We are to select 4 people for a committee, how manydifferent 4-people committees are possible?These are combinations so there are 13C4 possibilities.c.We are to select two men and two women for a 4-peoplecommittee, how many are possibilities?There are 2-steps to make the committee,
  45. 45. Permutations and CombinationsExample I. There are 5 men and 8 women.a. We are to select a president, a vice president, a manager,a treasurer from them. How many different possibilities arethere?These are permutations since the order is important.Hence there are 13P4 possibilities.b. We are to select 4 people for a committee, how manydifferent 4-people committees are possible?These are combinations so there are 13C4 possibilities.c.We are to select two men and two women for a 4-peoplecommittee, how many are possibilities?There are 2-steps to make the committee, select the2 women, then 2 men.
  46. 46. Permutations and CombinationsExample I. There are 5 men and 8 women.a. We are to select a president, a vice president, a manager, atreasurer from them. How many different possibilities arethere?These are permutations since the order is important.Hence there are 13P4 possibilities.b. We are to select 4 people for a committee, how manydifferent 4-people committees are possible?These are combinations so there are 13C4 possibilities.c.We are to select two men and two women for a 4-peoplecommittee, how many are possibilities?There are 2-steps to make the committee, select the2 women, then 2 men. There are 8C2 ways to select 2 women.
  47. 47. Permutations and CombinationsExample I. There are 5 men and 8 women.a. We are to select a president, a vice president, a manager, atreasurer from them. How many different possibilities arethere?These are permutations since the order is important.Hence there are 13P4 possibilities.b. We are to select 4 people for a committee, how manydifferent 4-people committees are possible?These are combinations so there are 13C4 possibilities.c.We are to select two men and two women for a 4-peoplecommittee, how many are possibilities?There are 2-steps to make the committee, select the2 women, then 2 men. There are 8C2 ways to select 2 women.There are 5C2 ways to select 2 men.
  48. 48. Permutations and CombinationsExample I. There are 5 men and 8 women.a. We are to select a president, a vice president, a manager, atreasurer from them. How many different possibilities arethere?These are permutations since the order is important.Hence there are 13P4 possibilities.b. We are to select 4 people for a committee, how manydifferent 4-people committees are possible?These are combinations so there are 13C4 possibilities.c.We are to select two men and two women for a 4-peoplecommittee, how many are possibilities?There are 2-steps to make the committee, select the2 women, then 2 men. There are 8C2 ways to select 2 women.There are 5C2 ways to select 2 men. Hence there are8C2 x 5C2 ways for to select the 2-men-2-women committees.
  49. 49. Permutations and Combinations

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