1. In SuperLotto Plus, a California state lottery game, you select five distinct numbers from 1 to 47, and one MEGA number from 1 to 27, hoping that your selection will match a random list selected by lottery officials. (a) How many different sets of six numbers can you select?

2. In SuperLotto Plus, a California state lottery game, you select five distinct numbers from 1 to 47, and one MEGA number from 1 to 27, hoping that your selection will match a random list selected by lottery officials. (a) How many different sets of six numbers can you select?
Answer: C(47,5)xC(27,1)
= 1,533,939 x 27
= 41,416,353

3. In SuperLotto Plus, a California state lottery game, you select five distinct numbers from 1 to 47, and one MEGA number from 1 to 27, hoping that your selection will match a random list selected by lottery officials. Eileen Burke always includes her age and her husband’s age as two of the first five numbers in her SuperLotto Plus selections. How many ways can she complete her list of six numbers?

4. In SuperLotto Plus, a California state lottery game, you select five distinct numbers from 1 to 47, and one MEGA number from 1 to 27, hoping that your selection will match a random list selected by lottery officials. Eileen Burke always includes her age and her husband’s age as two of the first five numbers in her SuperLotto Plus selections. How many ways can she complete her list of six numbers?
Answer: C(45,3)xC(27,1)
= 14,190x 27
= 383,130

5. Drawing Cards - How many cards must be drawn (without replacement) from a standard deck of 52 to guarantee drawing two cards of the same suit?

6. Drawing Cards - How many cards must be drawn (without replacement) from a standard deck of 52 to guarantee drawing two cards of the same suit?
Answer: 5

7. Drawing Cards - How many cards must be drawn (without replacement) from a standard deck of 52 to guarantee drawing three cards of the same suit?

8. Drawing Cards - How many cards must be drawn (without replacement) from a standard deck of 52 to guarantee drawing three cards of the same suit?
Answer: 9

9. How many different 5-card poker hands would contain only cards of a single suit?

10. How many different 5-card poker hands would contain only cards of a single suit?
Answer: 4 x C(13,5)
= 5148

11. Subject identification numbers in a certain scientific research project consist of three letters followed by three digits and then three more letters. Assume repetitions are not allowed within any of the three groups, but letters in the first group of three may occur also in the last group of three. How many distinct identification numbers are possible?

12. Subject identification numbers in a certain scientific research project consist of three letters followed by three digits and then three more letters. Assume repetitions are not allowed within any of the three groups, but letters in the first group of three may occur also in the last group of three. How many distinct identification numbers are possible?
Answer: P(26,3) x P(10,3) x P(26,3)
= 175,219,200,000

13. Radio stations in the United States have call letters that begin with K or W (for west or east of the Mississippi River, respectively). Some have three call letters, such as WBZ in Boston, WLS in Chicago, and KGO in San Francisco. Assuming no repetition of letters, how many three-letter sets of call letters are possible?

14. Radio stations in the United States have call letters that begin with K or W (for west or east of the Mississippi River, respectively). Some have three call letters, such as WBZ in Boston, WLS in Chicago, and KGO in San Francisco. Assuming no repetition of letters, how many three-letter sets of call letters are possible?
Answer: 2 x P(25,2)
= 1200

15. Most stations that were licensed after 1927 have four call letters starting with K or W, such as WXYZ in Detroit or KRLD in Dallas. Assuming no repetitions, how many four-letter sets are possible?

16. Most stations that were licensed after 1927 have four call letters starting with K or W, such as WXYZ in Detroit or KRLD in Dallas. Assuming no repetitions, how many four-letter sets are possible?
Answer: 2 x P(25,3)
= 27,600

17. Each team in an eight-team basketball league is scheduled to play each other team three times. How many games will be played altogether?

18. Each team in an eight-team basketball league is scheduled to play each other team three times. How many games will be played altogether?
Answer: 3 x C(8,2)
= 84

19. The Coyotes, a youth league baseball team, have seven pitchers, who only pitch, and twelve other players, all of whom can play any position other than pitcher. For Saturday’s game, the coach has not yet determined which nine players to use nor what the batting order will be, except that the pitcher will bat last. How many different batting orders may occur?

20. The Coyotes, a youth league baseball team, have seven pitchers, who only pitch, and twelve other players, all of whom can play any position other than pitcher. For Saturday’s game, the coach has not yet determined which nine players to use nor what the batting order will be, except that the pitcher will bat last. How many different batting orders may occur?
Answer: 7 x P(12,8)
= 139,708,800

21. A music class of eight girls and seven boys is having a recital
A music class of eight girls and seven boys is having a recital. If each member is to perform once, how many ways can the program be arranged so that a girl always performs first?

22. A music class of eight girls and seven boys is having a recital
A music class of eight girls and seven boys is having a recital. If each member is to perform once, how many ways can the program be arranged so that a girl always performs first?
Answer: 8 x 14!
= 697,426,329,600

23. A music class of eight girls and seven boys is having a recital
A music class of eight girls and seven boys is having a recital. If each member is to perform once, how many ways can the program be arranged so that a girl always performs first and a boy always performs second?

24. A music class of eight girls and seven boys is having a recital
A music class of eight girls and seven boys is having a recital. If each member is to perform once, how many ways can the program be arranged so that a girl always performs first and a boy always performs second?
Answer: 8 x 7 x 13!
= 348,713,164,800

25. A music class of eight girls and seven boys is having a recital
A music class of eight girls and seven boys is having a recital. If each member is to perform once, how many ways can the program be arranged so that Lisa always performs first and Doug always performs second?

26. A music class of eight girls and seven boys is having a recital
A music class of eight girls and seven boys is having a recital. If each member is to perform once, how many ways can the program be arranged so that Lisa always performs first and Doug always performs second?
Answer: 1x13!x1
= 6,227,020,800

27. A music class of eight girls and seven boys is having a recital
A music class of eight girls and seven boys is having a recital. If each member is to perform once, how many ways can the program be arranged so that the entire program will alternate between girls and boys?

28. A music class of eight girls and seven boys is having a recital
A music class of eight girls and seven boys is having a recital. If each member is to perform once, how many ways can the program be arranged so that the entire program will alternate between girls and boys?
Answer: 8! x 7!
= 203,212,800

29. A music class of eight girls and seven boys is having a recital
A music class of eight girls and seven boys is having a recital. If each member is to perform once, how many ways can the program be arranged so that the first, eighth and fifteenth performers must be girls?

30. A music class of eight girls and seven boys is having a recital
A music class of eight girls and seven boys is having a recital. If each member is to perform once, how many ways can the program be arranged so that the first, eighth and fifteenth performers must be girls?
Answer: 8 x 7 x 6 x 12!
= 160,944,537,600

31. Carole begins each day by reading from one of seven inspirational books. How many ways can she choose the books for one week if the selection is done by placing the book back on the bookshelf after she reads it?

32. Carole begins each day by reading from one of seven inspirational books. How many ways can she choose the books for one week if the selection is done by placing the book back on the bookshelf after she reads it?
Answer: 77
= 823,543

33. Carole begins each day by reading from one of seven inspirational books. How many ways can she choose the books for one week if the selection is done by choosing a different book each day?

34. Carole begins each day by reading from one of seven inspirational books. How many ways can she choose the books for one week if the selection is done by choosing a different book each day?
Answer: 7!
= 5,040

35. How many of the possible 5-card hands from a standard 52-card deck would consist of four clubs and one non-club?

36. How many of the possible 5-card hands from a standard 52-card deck would consist of four clubs and one non-club?
Answer: C(13,4) x 39
= 27,885

37. How many of the possible 5-card hands from a standard 52-card deck would consist of two face cards and three non-face cards?

38. How many of the possible 5-card hands from a standard 52-card deck would consist of two face cards and three non-face cards?
Answer: C(12, 2) x C(40, 3)
= 652, 080

39. How many of the possible 5-card hands from a standard 52-card deck would consist of two red cards two clubs and a spade?

40. How many of the possible 5-card hands from a standard 52-card deck would consist of two red cards two clubs and a spade?
Answer: C(26, 2) x C(13, 2) x 13
= 329,550

41. In how many ways could twenty-five people be divided into five groups containing, respectively, three, four, five, six, and seven people?

42. Answer: C(25,3) x C(22,4) x C(18,5) x C(13,6)
In how many ways could twenty-five people be divided into five groups containing, respectively, three, four, five, six, and seven people?
Answer: C(25,3) x C(22,4) x C(18,5) x C(13,6)
= x 1014

43. How many different three-number “combinations” are possible on a combination lock having 40 numbers on its dial? (Hint: “Combination” is a misleading name for these locks since repetitions are allowed and also order makes a difference.)

44. How many different three-number “combinations” are possible on a combination lock having 40 numbers on its dial? (Hint: “Combination” is a misleading name for these locks since repetitions are allowed and also order makes a difference.)
Answer : 403
= 64,000

45. Michael Grant, his wife and son, and four additional friends are driving, in two vehicles, to the seashore. If all seven people are available to drive, how many ways can the two drivers be selected? (Everyone would like to drive the sports car, so it is important which driver gets which car.)

46. Michael Grant, his wife and son, and four additional friends are driving, in two vehicles, to the seashore. If all seven people are available to drive, how many ways can the two drivers be selected? (Everyone would like to drive the sports car, so it is important which driver gets which car.)
Answer: P(7,2)
= 210

47. At the race track, you win the “daily double” by purchasing a ticket and selecting the winners of both of two specified races. If there are six and eight horses running in the first and second races, respectively, how many tickets must you purchase to guarantee a winning selection?

48. At the race track, you win the “daily double” by purchasing a ticket and selecting the winners of both of two specified races. If there are six and eight horses running in the first and second races, respectively, how many tickets must you purchase to guarantee a winning selection?
Answer: 6 x 8
= 48

49. Many race tracks offer a “trifecta” race
Many race tracks offer a “trifecta” race. You win by selecting the correct first-, second-, and third-place finishers. If eight horses are entered, how many tickets must you purchase to guarantee that one of them will be a trifecta winner?

50. Many race tracks offer a “trifecta” race
Many race tracks offer a “trifecta” race. You win by selecting the correct first-, second-, and third-place finishers. If eight horses are entered, how many tickets must you purchase to guarantee that one of them will be a trifecta winner?
Answer: P(8,3)
= 336

51. Because of his good work, Jeff Hubbard gets a contract to build homes on three additional blocks in the subdivision, with six homes on each block. He decides to build nine deluxe homes on these three blocks: two on the first block, three on the second, and four on the third. The remaining nine homes will be standard. Altogether on the three-block stretch, how many different choices does Jeff have for positioning the eighteen homes? (Hint: Consider the three blocks separately and use the fundamental counting principle.)

52. Because of his good work, Jeff Hubbard gets a contract to build homes on three additional blocks in the subdivision, with six homes on each block. He decides to build nine deluxe homes on these three blocks: two on the first block, three on the second, and four on the third. The remaining nine homes will be standard. Altogether on the three-block stretch, how many different choices does Jeff have for positioning the eighteen homes? (Hint: Consider the three blocks separately and use the fundamental counting principle.)
Answer: C(6,2) x C(6,3) x C(6,4)
= 4500

53. How many six-digit counting numbers can be formed using all six digits 4, 5, 6, 7, 8, and 9?

54. How many six-digit counting numbers can be formed using all six digits 4, 5, 6, 7, 8, and 9?
Answer: 6! or P(6,6)
= 720

55. A professor teaches a class of 60 students and another class of 40 students. Five percent of the students in each class are to receive a grade of A. How many different ways can the A grades be distributed?

56. A professor teaches a class of 60 students and another class of 40 students. Five percent of the students in each class are to receive a grade of A. How many different ways can the A grades be distributed?
Answer: C(60,3) x C(40,2)
= 26,691,600

57. How many counting numbers have four distinct nonzero digits such that the sum of the four digits is 12?

58. How many counting numbers have four distinct nonzero digits such that the sum of the four digits is 12?
Answer: 2 x 4!
= 48
Only 2 ways to get a sum of 12
Using 1,2,3 and 6 or 1,2,4 and 5

59. A computer company will screen a shipment of 30 processors by testing a random sample of five of them. How many different samples are possible?

60. A computer company will screen a shipment of 30 processors by testing a random sample of five of them. How many different samples are possible?
Answer: C(30,5)
= 142,506