/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 64 SENATE COMMITTEES In how many wa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 64

SENATE COMMITTEES In how many ways can a subcommittee of four be chosen from a Senate committee of five Democrats and four Republicans if a. All members are eligible? b. The subcommittee must consist of two Republicans and two Democrats?

Short Answer

Expert verified
In conclusion, there are \(126\) ways to form a subcommittee of \(4\) when all members are eligible, and there are \(60\) ways to form a subcommittee consisting of \(2\) Republicans and \(2\) Democrats.

Step by step solution

01

Scenario A: All members are eligible

To form a subcommittee of 4 from a group of 9 (5 Democrats and 4 Republicans), we can simply find the total number of combinations: \[C(9, 4) = \frac{9!}{4!(9-4)!}\] Calculate the combinations: \[C(9, 4) = \frac{9!}{4!5!} = \frac{362880}{(24)(120)} = 126\] So, there are 126 ways to form a subcommittee of 4 when all members are eligible.
02

Scenario B: Subcommittee must consist of 2 Republicans and 2 Democrats

Now, we need to create a subcommittee with exactly 2 Republicans and 2 Democrats. We will find the combinations separately for each group and then multiply the results to get the total number of ways. Find the number of ways to select 2 Republicans from the 4 available: \[C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{24}{(2)(2)} = 6\] Find the number of ways to select 2 Democrats from the 5 available: \[C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{120}{(2)(6)} = 10\] Now, multiply the combinations of Republicans and Democrats to get the total number of ways: Total ways = Ways to select Republicans * Ways to select Democrats Total ways = 6 * 10 = 60 So, there are 60 ways to form a subcommittee with 2 Republicans and 2 Democrats. In conclusion, there are 126 ways to form a subcommittee of 4 when all members are eligible, and there are 60 ways to form a subcommittee consisting of 2 Republicans and 2 Democrats.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Senate Committees
Senate committees play a crucial role in the legislative process. Imagine a large committee tasked with various political and administrative responsibilities.
These committees comprise senators from different political parties. The diversity of party members facilitates balanced decision-making.
  • Each committee can form smaller groups, known as subcommittees, to handle specific tasks.
  • Subcommittees focus on particular areas, allowing committee members to specialize and work more efficiently.
  • This specialization ensures a more thorough examination of topics, leading to more informed decisions.
The concept of forming subcommittees is vital as it mirrors the diverse and collective nature of the Senate, ensuring representation from different factions and viewpoints during decision-making processes.
Subcommittee Formation
Subcommittee formation involves selecting a smaller group from a larger committee to focus on particular aspects of legislation.
In our example, we have a committee of 9 senators, comprising 5 Democrats and 4 Republicans. We need to form a subcommittee of 4 members. Let's explore how this can be done.
  • There are generally multiple ways to form a subcommittee. Every member of the main committee is a potential candidate.
  • The selection criteria can vary. For instance, subcommittees can have open membership, or they can be formed with specific quotas, such as requiring certain numbers of members from each party.
  • In some cases, like our exercise, the subcommittee can be formed with restrictions, such as exact numbers of representatives from different groups, ensuring balanced participation.
The process of forming a subcommittee involves mathematical calculations to determine the number of potential combinations, ensuring fair representation based on the given criteria.
Combination Formula
The combination formula is a fundamental tool in combinatorics, used to calculate the number of ways to select items from a larger set.
This formula is represented as \[C(n, k) = \frac{n!}{k!(n-k)!}\]where \(n\) is the total number of items, \(k\) is the number of items to select, and \(!\) indicates factorial, a product of all positive integers up to that number.
Applying this formula helps us solve problems related to subcommittee formation by determining all possible selection combinations.
  • For instance, selecting 4 members from a group of 9 involves calculating \(C(9, 4)\).
  • This calculation includes dividing \(9!\), or 9 factorial, by the product of \(4!\) (the number of selected members) and \(5!\) (remaining members after selection).
  • Similarly, selecting 2 Republicans from 4 uses \(C(4, 2)\), while selecting 2 Democrats from 5 uses \(C(5, 2)\).

    Through these specific choice methods, we guarantee that each formation scenario, like different political party requirements, is thoroughly considered and accurately calculated.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In the opinion poll of Exercise 38, the voters were also asked to indicate their political affiliations-Democrat, Republican, or Independent. As before, let the letters \(L, M\), and \(U\) represent the low-, middle-, and upper-income groups, respectively. Let the letters \(D, R\) and \(I\) represent Democrat, Republican, and Independent, respectively. a. Describe a sample space corresponding to this poll. b. Describe the event \(E_{1}\) that a respondent is a Democrat. c. Describe the event \(E_{2}\) that a respondent belongs to the upper-income group and is a Republican. d. Describe the event \(E_{3}\) that a respondent belongs to the middle-income group and is not a Democrat.

In an online survey of 1962 executives from 64 countries conducted by Korn/Ferry International between August and October 2006 , the executives were asked if they would try to influence their children's career choices. Their replies: A (to a very great extent), \(\mathrm{B}\) (to a great extent), \(\mathrm{C}\) (to some extent), D (to a small extent), and \(\mathrm{E}\) (not at all) are recorded below: $$ \begin{array}{lccccc} \hline \text { Answer } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Respondents } & 135 & 404 & 1057 & 211 & 155 \\ \hline \end{array} $$ What is the probability that a randomly selected respondent's answer was \(\mathrm{D}\) (to a small extent) or \(\mathrm{E}\) (not at all)?

According to a survey of 176 retailers, \(46 \%\) of them use electronic tags as protection against shoplifting and employee theft. If one of these retailers is selected at random, what is the probability that he or she uses electronic tags as antitheft devices?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The numbers 1,2, and 3 are written separately on three pieces of paper. These slips of paper are then placed in a bowl. If you draw two slips from the bowl, one at a time and without replacement, then the sample space for this experiment consists of six elements.

Fifty raffle tickets are numbered 1 through 50 , and one of them is drawn at random. What is the probability that the number is a multiple of 5 or 7 ? Consider the following "solution": Since 10 tickets bear numbers that are multiples of 5 and since 7 tickets bear numbers that are multiples of 7 , we conclude that the required probability is $$ \frac{10}{50}+\frac{7}{50}=\frac{17}{50} $$ What is wrong with this argument? What is the correct answer?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.