A student council consists of 15 students. (a) How many ways can a committee of eight 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 6435 (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 eight 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 eight that contains A and not B is 1716 The number of ways to select a committee of eight that contains B and not A is 1287 The number of ways to select a committee of eight that contains neither A nor B is The total number of committees of eight that can be selected from the membership of the council is the sum Thus, the answer is 3003 x . (d) Suppose the council contains eight men and seven women. x (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 eight be selected from the council membersh 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, th total number of committees of eight that can be selected from the membership of the council is of the number of committees with A and not B, B and not A, and neither A nor (ii) How many committees of six contain at least one woman? The number of committees of six that contain at least one woman is (1) 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 , and the number of ways to perform step 2 is The number of committees of six with three men and three women is the ---Select--- of the number of ways to perform steps 1 and 2. Thus, the answer is (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 [ and similar calculations can be made for selecting the representatives from the other three classes. Thus, number of committees of eight that contain two representatives from each class is the Select... of four numbers, namely