1097 11.6 Basics of Counting Theory Solve each problem involving combinations. See Examples 7 and 8. 53. Seminar Presenters A banker’s association has 40 members. If 6 members are selected at random to present a seminar, how many different groups of 6 are possible? 54. Financial Planners Four financial planners are to be selected from a group of 12 to participate in a special program. In how many ways can this be done? In how many ways can the group that will not participate be selected? 55. Apple Samples How many different samples of 4 apples can be drawn from a crate of 25 apples? 56. Apple Samples Suppose that in Exercise 55 there are 5 rotten apples in the crate. (a) How many samples of 3 could be drawn in which all 3 are rotten? (b) How many samples of 3 could be drawn in which there are 2 good apples and 1 rotten apple? 57. Hamburger Choices Howard’s Hamburger Barn sells hamburgers with cheese, relish, lettuce, tomato, mustard, and/or ketchup. (a) How many different hamburgers can be made that use any 4 of the extras? (b) How many different hamburgers can be made if one of the 4 extras must be cheese? 58. Card Combinations Five playing cards having the numbers 2, 3, 4, 5, and 6 are shuffled and 2 cards are then drawn. How many different 2-card hands are possible? 59. Marble Samples If a bag contains 15 marbles, how many samples of 2 marbles can be drawn from it? How many samples of 4 marbles can be drawn? 60. Marble Samples If a bag contains 3 yellow, 4 white, and 8 blue marbles, how many samples of 2 can be drawn in which both marbles are blue? 61. Convention Delegation Choices A city council is composed of 5 liberals and 4 conservatives. Three members are to be selected randomly as delegates to a convention. (a) How many delegations are possible? (b) How many delegations could have all liberals? (c) How many delegations could have 2 liberals and 1 conservative? (d) If 1 member of the council serves as mayor, how many delegations are possible that include the mayor? 62. Delegation Choices Seven workers decide to send a delegation of 2 to their supervisor to discuss their grievances. (a) How many different delegations are possible? (b) If it is decided that a certain employee must be in the delegation, how many different delegations are possible? (c) If there are 2 women and 5 men in the group, how many delegations would include at least 1 woman? Use any method described in this section to solve each problem. See Examples 1–9. 63. Course Schedule Arrangement If Dwight has 8 courses to choose from, how many ways can he arrange his schedule if he must pick 4 of them? 64. Pineapple Samples How many samples of 9 pineapples can be drawn from a crate of 12?
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