Problem 31 - Algebra and Trigonometry

Question

A senate subcommittee consists of ten Democrats and seven Republicans. In how many ways can a chairman, vice chairman, and secretary be chosen if the chairman must be a Democrat and the vice chairman must be a Republican?

Step-by-Step Solution

Answer

There are 1050 ways to choose the officials.

Step 1: Choose the Chairman

To choose the chairman, who must be a Democrat, we have 10 candidates. So, there are 10 ways to choose the chairman.

Step 2: Choose the Vice Chairman

The vice chairman must be a Republican. There are 7 Republicans, so there are 7 ways to choose the vice chairman.

Step 3: Choose the Secretary

After selecting the chairman and vice chairman, there are 15 remaining members (since one Democrat and one Republican are already selected). There are no restrictions on the secretary, so we have 15 choices.

Step 4: Multiply the Choices

To find the total number of ways to select a chairman, vice chairman, and secretary, multiply the number of choices for each position: \[10 \times 7 \times 15 = 1050.\]

Key Concepts

Permutations

Permutations are a fundamental concept in combinatorics that help us determine the number of different ways to arrange a set of items where order matters. Imagine you have a row of chairs and a group of people to seat. The different possible line-ups of these people in the chairs are permutations. In mathematical terms, if you have a set of \( n \) items and you want to select \( r \) items in a specific order, the number of permutations is given by the formula:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

"!" denotes a factorial, meaning you multiply the number by every integer less than itself down to one.
For example, in the original exercise, once the chairman and vice chairman are selected, the remaining members are considered for the secretary position. This scenario requires us to consider the permutations of members, as the selection involves a specific ordering (chairman, vice chairman, and secretary).
Remember, whenever the order of selection is important—even if just the moral authority as in titles, we're talking about permutations. It ensures all roles are filled with the right individuals in a distinct sequence.

Combinations

Combinations are about picking items from a set where the order does not matter. Unlike permutations where the sequence is critical, combinations just care about selection. When you pick a team of people regardless of who sits where, that is a combination. For any set of \( n \) items, selecting \( r \) items is calculated by:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \)

This formula helps you determine how many ways you can select \( r \) items from \( n \) available items.
In the exercise, if we had to select members for a simple committee without designated roles, combinations would be our go-to concept. But since we need to assign roles like chairman and vice chairman specifically, the order does matter, hence we use permutations.
If a problem asks for arranging people or objects in a specific order, it's permutations you need. But, for grouping without regard to order, look to combinations.

Committee Selection

Choosing who gets to participate in roles within a committee is a classic problem in combinatorics. Often, these selections are predicated on rules—like needing a specific mix of backgrounds or roles, as seen in the exercise.

• Assigning distinct roles such as chairman and vice chairman.
• Restrictions on who can fill certain roles based on criteria (e.g., a Democrat must be chairman in this scenario).
• Consideration of all remaining members for other roles, ensuring that conditions are satisfied.

The problem involves detailed analysis of the criteria to ensure each role's requirements are met. A structured approach is often required:

• Identify roles and necessary criteria.
• Determine potential candidates for each role.
• Consider the order of selection as it impacts later choices.

A systematic approach ensures all criteria are met and the correct permutations are accounted for, just like how we calculated the choices for chairman, vice chairman, and secretary in the exercise. Knowing the rules and correctly applying them helps in efficiently deciding how many valid committees can be formed.