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 Two council members have the same major and are not permitted to serve together on a committee. How many ways can 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 Bis 3003 The total number of committees of eight that can be selected from the membership of the council is the sum X X

Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
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Chapter14: Counting And Probability
Section14.1: Counting
Problem 79E: Dance Committee A school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5...
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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 3003
X
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.
(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 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 eight that can be selected from the membership of the council is
(i)
X
of the number of committees with A and not B, B and not A, and neither A nor B.
(ii) How many committees of six contain at least one woman?
The number of committees of six that contain at least one woman is
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 product
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
number of committees of eight that contain two representatives from each class is the product
and similar calculations can be made for selecting the representatives from the other three classes. Thus, the
of four numbers. namelv
Transcribed Image Text: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 3003 X 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. (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 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 eight that can be selected from the membership of the council is (i) X of the number of committees with A and not B, B and not A, and neither A nor B. (ii) How many committees of six contain at least one woman? The number of committees of six that contain at least one woman is 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 product 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 number of committees of eight that contain two representatives from each class is the product and similar calculations can be made for selecting the representatives from the other three classes. Thus, the of four numbers. namelv
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