A student council consists of 15 students. (a) How many ways can a committee of five be selected from the membership of the council? As in Example 9.5.4, since a committee chosen from the members of the council is a subset of the council, the number of ways to select the committee is 3003 (b) Two council members have the same major and are not permitted to serve together on a committee. How many ways can a committee of five be selected from the membership of the council? As in Example 9.5.6, let A and B be the two council members who have the same major. The number of ways to select a committee of five that contains A and not B is 715 The number of ways to select a committee of five that contains B and not A is 715 The number of ways to select a committee of five that contains neither A nor B is 1287 The total number of committees of five that can be selected from the membership of the council is the sum of the number of committees with A and not B, B and not A, and neither A nor B. Thus, the answer is 4719 (c) Two council members insist on serving on committees together. If they cannot serve together, they will not serve at all. How many ways can a committee of five be selected from the council membership? As in Example 9.5.5, let A and B be the two council members who insist on serving together or not at all. Then some committees will contain both A and B and others will contain neither A nor B. So, the total number of committees of five that can be selected from the membership of the council is 1573 (d) Suppose the council contains eight men and seven women. (i) How many committees of six contain three men and three women? As in Example 9.5.7a, think of forming a committee as a two-step process, where step 1 is to choose the men and step 2 is to choose the women. The number of ways to perform step 1 is 56 The number of committees of six with three men and three women is the product (ii) How many committees of six contain at least one woman? The number of committees of six that contain at least one woman is of the number of ways to perform steps 1 and 2. Thus, the answer is 1960 (e) Suppose the council consists of three freshmen, four sophomores, three juniors, and five seniors. How many committees of eight contain two representatives from each class? The number of ways to select two representatives from the three freshmen is ---Select--- ✓ of four numbers, namely and the number of ways to perform step 2 is 35 and similar calculations can be made for selecting the representatives from the other three classes. Thus, the number of committees of eight that contain two representatives from each class is the