Question

In how many ways can 4 people be lined up?

Short answer

24

Step-by-step solution

- Understand the problem

The goal is to determine the number of ways to arrange 4 people in a line. This is a permutation problem because the order in which the people are arranged matters.

- Determine the formula

Permutations of n distinct objects can be calculated using the factorial function, denoted as \(! n!\). For 4 people, we need to find \(! 4!\).

- Compute the factorial

Calculate \(! 4!\), which is \(! 4 \times 3 \times 2 \times 1 = 24\).

Key concepts

Factorial

In mathematics, a factorial is a function that multiplies a given number by every whole number less than it until 1. For example, the factorial of 4, denoted as \(!4!\), is calculated as \[4! = 4 \times 3 \times 2 \times 1 = 24.\]

Factorials are a fundamental part of permutations. They help us determine how many different ways we can arrange a set of objects. The factorial function grows very quickly, meaning that even for small numbers, the result can be quite large.

Here are few key points about factorial:

• The factorial of 0 is defined as 1.
• Factorial is only defined for non-negative integers.
• It is used extensively in combinatorics and probability.

Arrangement

Arrangement refers to the way in which objects or people are placed. When dealing with permutations, the order matters. For four people, placing them in different positions changes the arrangement, making it a new permutation.

In the context of the exercise, we are arranging 4 people in a line. Each different order of these 4 people counts as a unique arrangement.

Following this logic:

• Arrangement 1: Person A, Person B, Person C, Person D.
• Arrangement 2: Person A, Person B, Person D, Person C.
• And so forth until all permutations are counted.

This demonstrates that for n objects, there are many possible arrangements, specifically calculated using the factorial function.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations and permutations of objects. It includes the study of how objects can be selected and arranged.

Key concepts in combinatorics include:

• Permutations, where order matters.
• Combinations, where order does not matter.
• Factorials, as used in calculating permutations.

Combinatorics is a powerful tool for solving many different types of problems in mathematics, computer science, and statistical physics.

Permutations Formula

The permutations formula is used to find the number of different ways to arrange a set of objects where the order is important.

The formula for permutations of n distinct objects is: \[P(n) = n!\]

For example, the number of permutations of 4 people (n=4) is calculated as: \[P(4) = 4! = 24\]

This means there are 24 different ways to line up 4 people. In general, if you want to arrange n objects, you use the factorial of n to determine the number of permutations. This simple formula provides a quick way to find out the different possible arrangements.