924 CHAPTER 13 Counting and Probability 56. Football Teams A defensive football squad consists of 25 players. Of these, 10 are linemen, 10 are linebackers, and 5 are safeties. How many different teams of 5 linemen, 3 linebackers, and 3 safeties can be formed? 57. Baseball In Major League Baseball a designated hitter may be used to hit for the pitcher. How many batting orders is it possible for a manager to use? (There are 9 regular players on a team.) 58. Baseball Suppose a baseball league exists where the pitcher must bat. The pitcher usually bats ninth. If this is the case, how many batting orders is it possible for a manager to use? 59. Baseball Teams A baseball team has 15 members. Four of the players are pitchers, and the remaining 11 members can play any position. How many different teams of 9 players can be formed? 60. World Series In the World Series the American League team ( )A and the National League team ( )N play until one team wins four games. If the sequence of winners is designated by letters (for example, NAAAA means that the National League team won the first game and the American League won the next four), how many different sequences are possible? 61. Basketball Teams A basketball team has 6 players who play guard (2 of 5 starting positions). How many different teams are possible, assuming that the remaining 3 positions are filled and it is not possible to distinguish a left guard from a right guard? 62. Basketball Teams On a basketball team of 12 players, 2 play only center, 3 play only guard, and the rest play forward (5 players on a team: 2 forwards, 2 guards, and 1 center). How many different teams are possible, assuming that it is not possible to distinguish a left guard from a right guard or a left forward from a right forward? 63. Combination Locks A combination lock displays 50 numbers. To open it, you turn clockwise to the first number of the “combination,” then rotate counterclockwise to the second number, and then rotate clockwise to the third number. (a) How many different lock combinations are there if repetition of numbers is allowed? If repetition of numbers is not allowed? (b) Comment on the description of such a lock as a combination lock. 64. Challenge Problem Passwords Suppose a password must have at least 8 characters, but no more than 12 characters, made up of letters (without distinction for case) and digits. If the password must contain at least one letter and at least one digit, how many passwords are possible? 44. Possible Answers on a Multiple-choice Test How many arrangements of answers are possible in a multiple-choice test with 5 questions, each of which has 4 possible answers? 45. Arranging Books Five different mathematics books are to be arranged on a student’s desk. How many arrangements are possible? 46. Forming License Plate Numbers How many different license plate numbers can be made using 2 letters followed by 4 digits selected from the digits 0 through 9, if: (a) Letters and digits may be repeated? (b) Letters may be repeated, but digits may not be repeated? (c) Neither letters nor digits may be repeated? 47. Birthday Problem In how many ways can 2 people each have different birthdays? Assume that there are 365 days in a year. 48. Birthday Problem In how many ways can 5 people all have different birthdays? Assume that there are 365 days in a year. 49. Forming a Committee A student dance committee is to be formed consisting of 2 boys and 3 girls. If the membership is to be chosen from 4 boys and 8 girls, how many different committees are possible? 50. Forming a Committee The student relations committee of a college consists of 2 administrators, 3 faculty members, and 5 students. Four administrators, 8 faculty members, and 20 students are eligible to serve. How many different committees are possible? 51. Forming Words How many different 9-letter words (meaningful or not) can be formed from the letters in the word ECONOMICS? 52. Forming Words How many different 11-letter words (meaningful or not) can be formed from the letters in the word MATHEMATICS? 53. Selecting Objects An urn contains 7 white balls and 3 red balls. Three balls are selected. In how many ways can the 3 balls be drawn from the total of 10 balls: (a) If 2 balls are white and 1 is red? (b) If all 3 balls are white? (c) If all 3 balls are red? 54. Selecting Objects An urn contains 15 red balls and 10 white balls. Five balls are selected. In how many ways can the 5 balls be drawn from the total of 25 balls: (a) If all 5 balls are red? (b) If 3 balls are red and 2 are white? (c) If at least 4 are red balls? 55. Senate Committees The U.S. Senate has 100 members. Suppose that it is desired to place each senator on exactly 1 of 7 possible committees. The first committee has 22 members, the second has 13, the third has 10, the fourth has 5, the fifth has 16, and the sixth and seventh have 17 apiece. In how many ways can these committees be formed?
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