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- 1. Permutations and Combinations
- 2. Permutations and CombinationsA k-permutation is an ordered lineup of k objects.
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?
- 26. Permutations and CombinationsExample G. How many 2-combinations from the set {a, b, c}are there?n = 3, k = 2,
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Permutations and Combinations

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