1098 CHAPTER 11 Further Topics in Algebra 65. Soup Ingredients Velma specializes in making different vegetable soups with carrots, celery, beans, peas, mushrooms, and potatoes. How many different soups can she make with any 4 ingredients? 66. Secretary/Manager Assignments From a pool of 7 administrative assistants, 3 are selected to be assigned to 3 managers, 1 assistant to each manager. In how many ways can this be done? 67. Musical Chairs Seatings In a game of musical chairs, 13 children will sit in 12 chairs. (1 will be left out.) How many seating arrangements are possible? 68. Plant Samples In an experiment on plant hardiness, a researcher gathers 6 wheat plants, 3 barley plants, and 2 rye plants. She wishes to select 4 plants at random. (a) In how many ways can this be done? (b) In how many ways can this be done if exactly 2 wheat plants must be included? 69. Committee Choices In a club with 8 women and 11 men members, how many 5-member committees can be chosen that satisfy the following conditions? (a) All are women. (b) All are men. (c) There are 3 women and 2 men. (d) There are no more than 3 men. 70. Committee Choices From 10 names on a ballot, 4 will be elected to a political party committee. In how many ways can the committee of 4 be formed if each person will have a different responsibility? 71. Combination Lock A briefcase has 2 locks. The combination to each lock consists of a 3-digit number, where digits may be repeated. How many combinations are possible? (Hint: The word combination is a misnomer. Lock combinations are permutations where the arrangement of the numbers is important.) 72. Combination Lock A typical “combination” for a padlock consists of 3 numbers from 0 to 39. Find the number of “combinations” that are possible with this type of lock, if a number may be repeated. 73. Garage Door Openers The code for some garage door openers consists of 12 electrical switches that can be set to either 0 or 1 by the owner. With this type of opener, how many codes are possible? (Data from Promax.) 74. Random Numbers To win a game, a person must pick 4 numbers from 0 to 9 in the correct order. If a number can be repeated, how many ways are there to select the numbers? 75. Keys In how many distinguishable ways can 4 keys be put on a circular key ring? 76. Sitting at a Round Table In how many different ways can 8 people sit at a round table? Assume that “a different way” means that at least 1 person is sitting next to someone different. Prove each statement for positive integers n and r, with r … n. (Hint: Use the definitions of permutations and combinations.) 77. P1n, n - 12 = P1n, n2 78. P1n, 12 = n 79. P1n, 02 = 1 80. P1n, n2 = n! 81. C1n, n2 = 1 82. C1n, 02 = 1 83. C10, 02 = 1 84. C1n, n - 12 = n 85. C1n, n - r2 = C1n, r2 86. Explain why the restriction r … n is needed in the formulas for C1n, r2 and P1n, r2. 0 5 10 30 25 20 35 3 2 1 2 1 3 2 1 3
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