Question

Four students are running in an election for class representative on the student council. In how many different ways can the four names be listed on the ballot?

Short answer

There are 24 different ways to list the four names.

Step-by-step solution

Understand the Problem

Determine the number of different ways to arrange four students' names on the ballot for class representative.

Recognize the Permutation

Since the order in which the names are listed matters, this is a permutation problem. The formula for the permutation of n items is given by: P(n) = n!

Set Up the Equation

For four students, we need to find P(4). This means we need to calculate the factorial of 4: P(4) = 4!

Calculate the Factorial

Calculate the factorial of 4 by multiplying all positive integers up to 4: 4! = 4 × 3 × 2 × 1

Compute the Value

Perform the multiplication: 4! = 4 × 3 × 2 × 1 = 24

Key concepts

Factorial

A 'factorial' is a fundamental concept in mathematics represented by the symbol \(!\). It is the product of all positive integers up to a given number. For instance, the factorial of 4 (written as \(!4\)) is calculated as follows: \(!4 = 4 \times 3 \times 2 \times 1 = 24\). This operation is crucial in many areas of combinatorics and probability because it counts the number of ways to arrange a set number of objects.
It's important to remember:

• 0! = 1 (by definition)
• Factorials grow very fast
• Used in many formulas for counting permutations and combinations

Using factorials simplifies complex problems, making calculations manageable by breaking them into predictable steps.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects.
It includes various subtopics like permutations and combinations, which help in solving problems related to arranging and selecting objects.
Basic concepts in combinatorics include:

• Permutations: Arrangements where order matters
• Combinations: Selections where order doesn't matter

Combinatorial methods are widely used in fields like computer science, probability, and statistics. For instance, by applying combinatorial logic, we can determine how many ways we can list students on a ballot.

Permutations and Combinations

Permutations and combinations make up core parts of combinatorics. While they may sound similar, they handle different scenarios:

• Permutations: Focus on the arrangement of items (order matters). Example: Arranging 4 students is a permutation and calculated using factorials
• Combinations: Focus on selecting items (order doesn't matter). Example: Choosing 2 out of 4 students for a committee without regard to order

The difference is crucial to solving problems correctly. Our ballot example uses permutations, calculated as \(!4 = 24\), since the order of names significantly affects the outcome headcount.

Mathematical Calculations

Mathematical calculations often involve performing operations with precision. In the context of our problem:

• Understanding Permutation: Recognize it's about order
• Applying Factorial: Use the factorial formula for calculations

Steps for calculation:
1. Write down the factorial formula: \(!n\).
2. Substitute \(n\) with the given number: \(4!\).
3. Perform multiplication: \(4 \times 3 \times 2 \times 1\).
Thus, the total number of permutations of 4 students is 24. Accuracy in these calculations is essential for correct results.