Problem 64 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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?

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Key Concepts

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.