Problem 31 - Algebra and Trigonometry

Question

Class Executive In how many ways can a president, vice president, and secretary be chosen from a class of 30 students?

Step-by-Step Solution

Answer

There are 24,360 ways to choose the positions.

Step 1: Understand the Problem

We need to choose 3 specific positions (President, Vice President, Secretary) from a group of 30 students. This is a permutation problem because each position is distinct and order matters.

Step 2: Set up the Formula for Permutations

Permutations are calculated using the formula \( nPr = \frac{n!}{(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items chosen. Here, \( n = 30 \) and \( r = 3 \).

Step 3: Calculate the Permutation

Insert \( n = 30 \) and \( r = 3 \) into the formula: \[30P3 = \frac{30!}{(30-3)!} = \frac{30 \times 29 \times 28 \times 27!}{27!}\].The \( 27! \) cancels out, simplifying to \( 30 \times 29 \times 28 \).

Step 4: Perform the Multiplication

Calculate \( 30 \times 29 \times 28 = 30 \cdot 29 = 870 \, \text{and} \, 870 \cdot 28 = 24360 \). Therefore, there are 24,360 different ways to choose these 3 positions.

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics that focuses on counting, arranging, and grouping objects. In problems like selecting class executives, combinatorics helps us determine the number of possible arrangements or selections.

• Permutations: These are about arranging objects where order matters, just like selecting a president, vice president, and secretary.
• Combinations: These are used when the order of selection does not matter.

In our problem, we specifically look at permutations because each role is unique and has a specific order. Understanding these differences is key for solving a wide variety of counting problems.

Factorial

The factorial, denoted by an exclamation mark (!), is a product of an integer and all the integers below it. It is a foundational concept in permutations in combinatorics. For example, 5! (read as "five factorial") equals 5 × 4 × 3 × 2 × 1, which is 120.
Factorials are essential for calculating permutations because they help account for the different ways items can be ordered. In our problem with 30 students and 3 positions, we use factorials to express the total arrangements:- Total ways to arrange 30 students is 30!- We only need 3 positions, so we use the formula \[ nPr = \frac{n!}{(n-r)!} \]This formula utilizes factorials to efficiently calculate the number of specific arrangements, leading us to a more manageable computation.

Distinct Positions

In the given problem, the concept of distinct positions means that each role (president, vice president, secretary) is unique. This is a crucial aspect because it influences how we calculate the problem's solution.
Distinct positions imply that rearranging the three positions leads to a different outcome. For example:

• A student selected as president cannot be selected as a vice president in the same selection.
• Each position needs to be filled in a specific order with no overlap.

This uniqueness demands that we consider each position separately, using permutations rather than combinations, ensuring each choice reflects its own specific arrangement.

Order Matters

Order matters in permutations because rearranging items leads to different outcomes. In the context of our exercise, selecting a president, vice president, and secretary from 30 students requires us to focus on who holds each position.
Here’s why order is critical:

• The sequence of choosing individuals for each position affects which arrangement we deem different.
• If we swap the president and vice president roles, it creates a completely new lineup.

Ultimately, because order impacts the structure of arrangements so deeply, permutations come into play, giving a method to calculate possible configurations. Focusing on the order emphasizes the importance each role plays in the outcome.