74 permutations and combinations
- 2. Permutations and Combinations A k-permutation is an ordered lineup of k objects.
- 3. Permutations and Combinations A k-permutation is an ordered lineup of k objects. Example A. List all the 2-permutations taken from {a, b, c}.
- 4. Permutations and Combinations A 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. Permutations and Combinations A 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 Example B. List all the 3-permutations taken from {a, b, c}.
- 6. Permutations and Combinations A 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 Example B. List all the 3-permutations taken from {a, b, c}. abc, acb, bac, bca, cab, cba
- 7. Permutations and Combinations A 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 Example B. List all the 3-permutations taken from {a, b, c}. abc, acb, bac, bca, cab, cba The number of k-permutations (ordered arrangements) taken from n objects is: n! nPk = (n – k)!
- 8. Permutations and Combinations A 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 Example B. List all the 3-permutations taken from {a, b, c}. abc, acb, bac, bca, cab, cba The number of k-permutations (ordered arrangements) taken from n objects is: n! nPk = (n – k)! Example C. How many 2-permutations taken from {a, b, c} are there?
- 9. Permutations and Combinations A 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 Example B. List all the 3-permutations taken from {a, b, c}. abc, acb, bac, bca, cab, cba The number of k-permutations (ordered arrangements) taken from 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. Permutations and Combinations A 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 Example B. List all the 3-permutations taken from {a, b, c}. abc, acb, bac, bca, cab, cba The number of k-permutations (ordered arrangements) taken from 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. Permutations and Combinations A 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 Example B. List all the 3-permutations taken from {a, b, c}. abc, acb, bac, bca, cab, cba The number of k-permutations (ordered arrangements) taken from n objects is: n! nPk = (n – k)! Example C. How many 2-permutations taken from {a, b, c} are there? 3! 6 n=3, k=2, 3P2 = (3 – 2)! = 1 = 6
- 12. Permutations and Combinations A 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 Example B. List all the 3-permutations taken from {a, b, c}. abc, acb, bac, bca, cab, cba The number of k-permutations (ordered arrangements) taken from n objects is: n! nPk = (n – k)! Example C. How many 2-permutations taken from {a, b, c} are there? 3! 6 n=3, k=2, 3P2 = (3 – 2)! = 1 = 6 They are {ab, ba, ac, ca, bc, cb}.
- 13. Permutations and Combinations Example D. How many different arrangements of 7 people from a group of 10 people in a row of 7 seats are there?
- 14. Permutations and Combinations Example D. How many different arrangements of 7 people from a group of 10 people in a row of 7 seats are there? There are 10 people so n = 10.
- 15. Permutations and Combinations Example D. How many different arrangements of 7 people from 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 in order so k = 7,
- 16. Permutations and Combinations Example D. How many different arrangements of 7 people from 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 in order so k = 7, so there are 10P7 = 10! / (10 – 7)!
- 17. Permutations and Combinations Example D. How many different arrangements of 7 people from 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 in order so k = 7, so there are 10P7 = 10! / (10 – 7)! =10! / 3!
- 18. Permutations and Combinations Example D. How many different arrangements of 7 people from 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 in order so k = 7, so there are 10P7 = 10! / (10 – 7)! =10! / 3! = 10 x 9 x 8 x .. X 4 = 604800 possibilities.
- 19. Permutations and Combinations Example D. How many different arrangements of 7 people from 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 in order so k = 7, so there are 10P7 = 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. Permutations and Combinations Example D. How many different arrangements of 7 people from 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 in order so k = 7, so there are 10P7 = 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. Permutations and Combinations Example D. How many different arrangements of 7 people from 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 in order so k = 7, so there are 10P7 = 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. Permutations and Combinations Example D. How many different arrangements of 7 people from 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 in order so k = 7, so there are 10P7 = 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. Permutations and Combinations Example D. How many different arrangements of 7 people from 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 in order so k = 7, so there are 10P7 = 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. Permutations and Combinations Example D. How many different arrangements of 7 people from 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 in order so k = 7, so there are 10P7 = 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) taken from n objects is: n! nCk = (n – k)!k!
- 25. Permutations and Combinations Example G. How many 2-combinations from the set {a, b, c} are there?
- 26. Permutations and Combinations Example G. How many 2-combinations from the set {a, b, c} are there? n = 3, k = 2,
- 27. Permutations and Combinations Example 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. Permutations and Combinations Example G. How many 2-combinations from the set {a, b, c} are there? n = 3, k = 2, hence there are 3! 3 3C2 = (3 – 2)!2!
- 29. Permutations and Combinations Example G. How many 2-combinations from the set {a, b, c} are there? n = 3, k = 2, hence there are 3! 3 3C2 = =3 (3 – 2)!2!
- 30. Permutations and Combinations Example G. How many 2-combinations from the set {a, b, c} are there? n = 3, k = 2, hence there are 3! 3 3C2 = =3 (3 – 2)!2! So there are three 2-combinations.
- 31. Permutations and Combinations Example G. How many 2-combinations from the set {a, b, c} are there? n = 3, k = 2, hence there are 3! 3 3C2 = =3 (3 – 2)!2! So there are three 2-combinations. They are {a, b}, {a, c}, {b, c}.
- 32. Permutations and Combinations Example G. How many 2-combinations from the set {a, b, c} are there? n = 3, k = 2, hence there are 3! 3 3C2 = =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 chicken dish, a vegetable dish, fried rice, and fried noodles. We may chose any 3 of them for a 3-Combo Special. How many different 3-Combo Specials are possible?
- 33. Permutations and Combinations Example G. How many 2-combinations from the set {a, b, c} are there? n = 3, k = 2, hence there are 3! 3 3C2 = =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 chicken dish, a vegetable dish, fried rice, and fried noodles. We may chose any 3 of them for a 3-Combo Special. How many different 3-Combo Specials are possible? n = 5, we are to take 3 of so k = 3,
- 34. Permutations and Combinations Example G. How many 2-combinations from the set {a, b, c} are there? n = 3, k = 2, hence there are 3! 3 3C2 = =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 chicken dish, a vegetable dish, fried rice, and fried noodles. We may chose any 3 of them for a 3-Combo Special. How many different 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. Permutations and Combinations Example G. How many 2-combinations from the set {a, b, c} are there? n = 3, k = 2, hence there are 3! 3 3C2 = =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 chicken dish, a vegetable dish, fried rice, and fried noodles. We may chose any 3 of them for a 3-Combo Special. How many different 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. Permutations and Combinations Example G. How many 2-combinations from the set {a, b, c} are there? n = 3, k = 2, hence there are 3! 3 3C2 = =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 chicken dish, a vegetable dish, fried rice, and fried noodles. We may chose any 3 of them for a 3-Combo Special. How many different 3-Combo Specials are possible? n = 5, we are to take 3 of so k = 3, hence there are 5*4 5! 5! 5C3 = (5 – 3)!3! = 2!3!
- 37. Permutations and Combinations Example G. How many 2-combinations from the set {a, b, c} are there? n = 3, k = 2, hence there are 3! 3 3C2 = =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 chicken dish, a vegetable dish, fried rice, and fried noodles. We may chose any 3 of them for a 3-Combo Special. How many different 3-Combo Specials are possible? n = 5, we are to take 3 of so k = 3, hence there are 5*4 5! 5! 5C3 = (5 – 3)!3! = 2!3! = 10 3-Combos specials.
- 38. Permutations and Combinations Example 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 are there?
- 39. Permutations and Combinations Example 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 are there? These are permutations since the order is important.
- 40. Permutations and Combinations Example 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 are there? These are permutations since the order is important. Hence there are 13P4 possibilities.
- 41. Permutations and Combinations Example 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 are there? These are permutations since the order is important. Hence there are 13P4 possibilities. b. We are to select 4 people for a committee, how many different 4-people committees are possible?
- 42. Permutations and Combinations Example 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 are there? These are permutations since the order is important. Hence there are 13P4 possibilities. b. We are to select 4 people for a committee, how many different 4-people committees are possible? These are combinations so there are 13C4 possibilities.
- 43. Permutations and Combinations Example 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 are there? These are permutations since the order is important. Hence there are 13P4 possibilities. b. We are to select 4 people for a committee, how many different 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-people committee, how many are possibilities?
- 44. Permutations and Combinations Example 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 are there? These are permutations since the order is important. Hence there are 13P4 possibilities. b. We are to select 4 people for a committee, how many different 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-people committee, how many are possibilities? There are 2-steps to make the committee,
- 45. Permutations and Combinations Example 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 are there? These are permutations since the order is important. Hence there are 13P4 possibilities. b. We are to select 4 people for a committee, how many different 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-people committee, how many are possibilities? There are 2-steps to make the committee, select the 2 women, then 2 men.
- 46. Permutations and Combinations Example 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 are there? These are permutations since the order is important. Hence there are 13P4 possibilities. b. We are to select 4 people for a committee, how many different 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-people committee, how many are possibilities? There are 2-steps to make the committee, select the 2 women, then 2 men. There are 8C2 ways to select 2 women.
- 47. Permutations and Combinations Example 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 are there? These are permutations since the order is important. Hence there are 13P4 possibilities. b. We are to select 4 people for a committee, how many different 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-people committee, how many are possibilities? There are 2-steps to make the committee, select the 2 women, then 2 men. There are 8C2 ways to select 2 women. There are 5C2 ways to select 2 men.
- 48. Permutations and Combinations Example 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 are there? These are permutations since the order is important. Hence there are 13P4 possibilities. b. We are to select 4 people for a committee, how many different 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-people committee, how many are possibilities? There are 2-steps to make the committee, select the 2 women, then 2 men. There are 8C2 ways to select 2 women. There are 5C2 ways to select 2 men. Hence there are 8C2 x 5C2 ways for to select the 2-men-2-women committees.