11X1 T06 03 combinations (2010)

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1. Combinations 2. Combinations A combination is a set of objects where the order that they are arranged is not important. 3. Combinations A combination is a set of objects where the order that they are arranged is not important. If we arrange objects in a line, and the order is not important then; 4. Combinations A combination is a set of objects where the order that they are arranged is not important. If we arrange objects in a line, and the order is not important then; A B is the same arrangement as B A 5. Combinations A combination is a set of objects where the order that they are arranged is not important. If we arrange objects in a line, and the order is not important then; A B is the same arrangement as B A e.g. 5 objects, arrange 2 of them 6. Combinations A combination is a set of objects where the order that they are arranged is not important. If we arrange objects in a line, and the order is not important then; A B is the same arrangement as B A e.g. 5 objects, arrange 2 of them A B B A C A D A E A A C B C C B D B E B A D B D C D D C E C A E B E C E D E E D 7. Combinations A combination is a set of objects where the order that they are arranged is not important. If we arrange objects in a line, and the order is not important then; A B is the same arrangement as B A e.g. 5 objects, arrange 2 of them A B B A C A D A E A A C B C C B D B E B A D B D C D D C E C A E B E C E D E E D Permutations 5P2 20 8. Combinations A combination is a set of objects where the order that they are arranged is not important. If we arrange objects in a line, and the order is not important then; A B is the same arrangement as B A e.g. 5 objects, arrange 2 of them A B B A C A D A E A A C B C C B D B E B A D B D C D D C E C A E B E C E D E E D Permutations 5P2 20 9. Combinations A combination is a set of objects where the order that they are arranged is not important. If we arrange objects in a line, and the order is not important then; A B is the same arrangement as B A e.g. 5 objects, arrange 2 of them A B B A C A D A E A A C B C C B D B E B A D B D C D D C E C A E B E C E D E E D 20 Permutations 5P2 Combinations 2! 20 10 10. 5 objects, arrange 3 of them 11. 5 objects, arrange 3 of them A B C B A C C A B D A B E A B A B D B A D C A D D A C E A C A B E B A E C A E D A E E A D A C B B C A C B A D B A E B A A C D B C D C B D D B C E B C A C E B C E C B E D B E E B D A D B B D A C D A D C A E C A A D C B D C C D B D C B E C B A D E B D E C D E D C E E C D A E B B E A C E A D E A E D A A E C B E C C E B D E B E D B A E D B E D C E D D E C E D C 12. 5 objects, arrange 3 of them A B C B A C C A B D A B E A B A B D B A D C A D D A C E A C A B E B A E C A E D A E E A D A C B B C A C B A D B A E B A A C D B C D C B D D B C E B C A C E B C E C B E D B E E B D A D B B D A C D A D C A E C A A D C B D C C D B D C B E C B A D E B D E C D E D C E E C D A E B B E A C E A D E A E D A A E C B E C C E B D E B E D B A E D B E D C E D D E C E D C Permutations 5P3 60 13. 5 objects, arrange 3 of them A B C B A C C A B D A B E A B A B D B A D C A D D A C E A C A B E B A E C A E D A E E A D A C B B C A C B A D B A E B A A C D B C D C B D D B C E B C A C E B C E C B E D B E E B D A D B B D A C D A D C A E C A A D C B D C C D B D C B E C B A D E B D E C D E D C E E C D A E B B E A C E A D E A E D A A E C B E C C E B D E B E D B A E D B E D C E D D E C E D C Permutations 5P3 60 14. 5 objects, arrange 3 of them A B C B A C C A B D A B E A B A B D B A D C A D D A C E A C A B E B A E C A E D A E E A D A C B B C A C B A D B A E B A A C D B C D C B D D B C E B C A C E B C E C B E D B E E B D A D B B D A C D A D C A E C A A D C B D C C D B D C B E C B A D E B D E C D E D C E E C D A E B B E A C E A D E A E D A A E C B E C C E B D E B E D B A E D B E D C E D D E C E D C 60 Permutations 5P3 Combinations 3! 60 10 15. If we have n different objects, and we arrange k of them and are not concerned about the order; 16. If we have n different objects, and we arrange k of them and are not concerned about the order; n Pk Number of Arrangements k! 17. If we have n different objects, and we arrange k of them and are not concerned about the order; n Pk Number of Arrangements k! n! n k !k! 18. If we have n different objects, and we arrange k of them and are not concerned about the order; n Pk Number of Arrangements k! n! n k !k! nCk 19. If we have n different objects, and we arrange k of them and are not concerned about the order; n Pk Number of Arrangements k! n! n k !k! nCk e.g. (i) How many ways can 6 numbers be chosen from 45 numbers? 20. If we have n different objects, and we arrange k of them and are not concerned about the order; n Pk Number of Arrangements k! n! n k !k! nCk e.g. (i) How many ways can 6 numbers be chosen from 45 numbers? Ways 45C6 21. If we have n different objects, and we arrange k of them and are not concerned about the order; n Pk Number of Arrangements k! n! n k !k! nCk e.g. (i) How many ways can 6 numbers be chosen from 45 numbers? Ways 45C6 8145060 22. If we have n different objects, and we arrange k of them and are not concerned about the order; n Pk Number of Arrangements k! n! n k !k! nCk e.g. (i) How many ways can 6 numbers be chosen from 45 numbers? Ways 45C6 8145060 Note: at 40 cents per game, $3 258 024 = amount of money you have to spend to guarantee a win in Lotto. 23. (ii) Committees of five people are to be obtained from a group of seven men and four women. How many committees are possible if; a) there are no restrictions? 24. (ii) Committees of five people are to be obtained from a group of seven men and four women. How many committees are possible if; a) there are no restrictions? Committees 11C5 25. (ii) Committees of five people are to be obtained from a group of seven men and four women. How many committees are possible if; a) there are no restrictions? Committees 11C5 With no restrictions, choose 5 people from 11, gender does not matter 26. (ii) Committees of five people are to be obtained from a group of seven men and four women. How many committees are possible if; a) there are no restrictions? Committees 11C5 With no restrictions, choose 5 people 462 from 11, gender does not matter 27. (ii) Committees of five people are to be obtained from a group of seven men and four women. How many committees are possible if; a) there are no restrictions? Committees 11C5 With no restrictions, choose 5 people 462 from 11, gender does not matter b) the committee contains only males? 28. (ii) Committees of five people are to be obtained from a group of seven men and four women. How many committees are possible if; a) there are no restrictions? Committees 11C5 With no restrictions, choose 5 people 462 from 11, gender does not matter b) the committee contains only males? Committees 7C5 29. (ii) Committees of five people are to be obtained from a group of seven men and four women. How many committees are possible if; a) there are no restrictions? Committees 11C5 With no restrictions, choose 5 people 462 from 11, gender does not matter b) the committee contains only males? Committees 7C5 By restricting it to only males, there is only 7 people to choose from 30. (ii) Committees of five people are to be obtained from a group of seven men and four women. How many committees are possible if; a) there are no restrictions? Committees 11C5 With no restrictions, choose 5 people 462 from 11, gender does not matter b) the committee contains only males? Committees 7C5 By restricting it to only males, there is 21 only 7 people to choose from 31. (ii) Committees of five people are to be obtained from a group of seven men and four women. How many committees are possible if; a) there are no restrictions? Committees 11C5 With no restrictions, choose 5 people 462 from 11, gender does not matter b) the committee contains only males? Committees 7C5 By restricting it to only males, there is 21 only 7 people to choose from c) the committee contains at least one woman? 32. (ii) Committees of five people are to be obtained from a group of seven men and four women. How many committees are possible if; a) there are no restrictions? Committees 11C5 With no restrictions, choose 5 people 462 from 11, gender does not matter b) the committee contains only males? Committees 7C5 By restricting it to only males, there is 21 only 7 people to choose from c) the committee contains at least one woman? Committees 462 21 33. (ii) Committees of five people are to be obtained from a group of seven men and four women. How many committees are possible if; a) there are no restrictions? Committees 11C5 With no restrictions, choose 5 people 462 from 11, gender does not matter b) the committee contains only males? Committees 7C5 By restricting it to only males, there is 21 only 7 people to choose from c) the committee contains at least one woman? Committees 462 21 easier to work out male only and subtract from total number of committees 34. (ii) Committees of five people are to be obtained from a group of seven men and four women. How many committees are possible if; a) there are no restrictions? Committees 11C5 With no restrictions, choose 5 people 462 from 11, gender does not matter b) the committee contains only males? Committees 7C5 By restricting it to only males, there is 21 only 7 people to choose from c) the committee contains at least one woman? Committees 462 21 easier to work out male only and subtract 441 from total number of committees 35. (iii) A hand of five cards is dealt from a regular pack of fifty two cards. a) What is the number of possible hands? 36. (iii) A hand of five cards is dealt from a regular pack of fifty two cards. a) What is the number of possible hands? Hands 52C5 37. (iii) A hand of five cards is dealt from a regular pack of fifty two cards. a) What is the number of possible hands? Hands 52C5 2598960 38. (iii) A hand of five cards is dealt from a regular pack of fifty two cards. a) What is the number of possible hands? Hands 52C5 2598960 b) What is the probability of getting three of a kind? 39. (iii) A hand of five cards is dealt from a regular pack of fifty two cards. a) What is the number of possible hands? Hands 52C5 2598960 b) What is the probability of getting three of a kind? choose which number has " three of a kind" Hands 13C1 40. (iii) A hand of five cards is dealt from a regular pack of fifty two cards. a) What is the number of possible hands? Hands 52C5 2598960 b) What is the probability of getting three of a kind? choose which number has choose three of " three of a kind" those cards Hands 13C14 C3 41. (iii) A hand of five cards is dealt from a regular pack of fifty two cards. a) What is the number of possible hands? Hands 52C5 2598960 b) What is the probability of getting three of a kind? choose which number has choose three of " three of a kind" those cards choose remaining Hands 13C14 C3 48 C2 two cards from the rest 42. (iii) A hand of five cards is dealt from a regular pack of fifty two cards. a) What is the number of possible hands? Hands 52C5 2598960 b) What is the probability of getting three of a kind? choose which number has choose three of " three of a kind" those cards choose remaining Hands 13C14 C3 48 C2 two cards from the rest 58656 43. (iii) A hand of five cards is dealt from a regular pack of fifty two cards. a) What is the number of possible hands? Hands 52C5 2598960 b) What is the probability of getting three of a kind? choose which number has choose three of " three of a kind" those cards choose remaining Hands 13C14 C3 48 C2 two cards from the rest 58656 58656 Pthree of a kind 2598960 44. (iii) A hand of five cards is dealt from a regular pack of fifty two cards. a) What is the number of possible hands? Hands 52C5 2598960 b) What is the probability of getting three of a kind? choose which number has choose three of " three of a kind" those cards choose remaining Hands 13C14 C3 48 C2 two cards from the rest 58656 58656 Pthree of a kind 2598960 94 2915 45. (iii) A hand of five cards is dealt from a regular pack of fifty two cards. a) What is the number of possible hands? Hands 52C5 2598960 b) What is the probability of getting three of a kind? choose which number has choose three of " three of a kind" those cards choose remaining Hands 13C14 C3 48 C2 two cards from the rest 58656 58656 Pthree of a kind 2598960 94 (=3.2%) 2915 46. 2004 Extension 1 HSC Q2e) A four person team is to be chosen at random from nine women and seven men. (i) In how many ways can this team be chosen? 47. 2004 Extension 1 HSC Q2e) A four person team is to be chosen at random from nine women and seven men. (i) In how many ways can this team be chosen? Teams 16C4 48. 2004 Extension 1 HSC Q2e) A four person team is to be chosen at random from nine women and seven men. (i) In how many ways can this team be chosen? Teams 16C4 With no restrictions, choose 4 people from 16, gender does not matter 49. 2004 Extension 1 HSC Q2e) A four person team is to be chosen at random from nine women and seven men. (i) In how many ways can this team be chosen? Teams 16C4 With no restrictions, choose 4 people 1820 from 16, gender does not matter 50. 2004 Extension 1 HSC Q2e) A four person team is to be chosen at random from nine women and seven men. (i) In how many ways can this team be chosen? Teams 16C4 With no restrictions, choose 4 people 1820 from 16, gender does not matter (ii) What is the probability that the team will consist of four women? 51. 2004 Extension 1 HSC Q2e) A four person team is to be chosen at random from nine women and seven men. (i) In how many ways can this team be chosen? Teams 16C4 With no restrictions, choose 4 people 1820 from 16, gender does not matter (ii) What is the probability that the team will consist of four women? Teams 9C4 52. 2004 Extension 1 HSC Q2e) A four person team is to be chosen at random from nine women and seven men. (i) In how many ways can this team be chosen? Teams 16C4 With no restrictions, choose 4 people 1820 from 16, gender does not matter (ii) What is the probability that the team will consist of four women? Teams 9C4 By restricting it to only women, there is only 9 people to choose from 53. 2004 Extension 1 HSC Q2e) A four person team is to be chosen at random from nine women and seven men. (i) In how many ways can this team be chosen? Teams 16C4 With no restrictions, choose 4 people 1820 from 16, gender does not matter (ii) What is the probability that the team will consist of four women? Teams 9C4 By restricting it to only women, there is 126 only 9 people to choose from 54. 2004 Extension 1 HSC Q2e) A four person team is to be chosen at random from nine women and seven men. (i) In how many ways can this team be chosen? Teams 16C4 With no restrictions, choose 4 people 1820 from 16, gender does not matter (ii) What is the probability that the team will consist of four women? Teams 9C4 By restricting it to only women, there is 126 only 9 people to choose from 126 P4 women team 1820 55. 2004 Extension 1 HSC Q2e) A four person team is to be chosen at random from nine women and seven men. (i) In how many ways can this team be chosen? Teams 16C4 With no restrictions, choose 4 people 1820 from 16, gender does not matter (ii) What is the probability that the team will consist of four women? Teams 9C4 By restricting it to only women, there is 126 only 9 people to choose from 126 P4 women team 1820 9 130 56. 2004 Extension 1 HSC Q2e) A four person team is to be chosen at random from nine women and seven men. (i) In how many ways can this team be chosen? Teams 16C4 With no restrictions, choose 4 people 1820 from 16, gender does not matter (ii) What is the probability that the team will consist of four women? Teams 9C4 By restricting it to only women, there is 126 only 9 people to choose from 126 P4 women team 1820 Exercise 10G; odd 9 (not 19, 27) 130