Problem 60 - Elementary Algebra

Question

The number of different ways 5 people can be arranged in a row is \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 .\) How many ways is this?

Step-by-Step Solution

Answer

Answer: 120

Step 1: Identify the formula for permutation

We will solve this problem using the factorial formula, with n being the number of objects to arrange (in our case, 5 people). The formula for calculating the number of permutations is: n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1

Step 2: Substitute the number of objects in the formula

In our problem, we have 5 people, so n = 5. We will substitute this value into the formula we found in Step 1: 5! = 5 × (5-1) × (5-2) × ... × 3 × 2 × 1

Step 3: Solve the factorial expression

Now we will evaluate the expression by multiplying each term: 5! = 5 × 4 × 3 × 2 × 1 = 120

Step 4: Interpret the result

The result of 120 represents the number of different ways 5 people can be arranged in a row. So there are 120 possible arrangements for these 5 people.

Key Concepts

Factorial

The factorial of a number is a fundamental concept in mathematics, especially in permutations and combinations. It is represented by an exclamation mark (!), for example, "5!", and it means to multiply the number by all positive whole numbers less than itself.

To find the factorial of 5, which can be written as 5!, you would calculate:

• 5 × 4 = 20
• 20 × 3 = 60
• 60 × 2 = 120
• 120 × 1 = 120

This gives us that 5! = 120. When you see a factorial, always remember it is multiplying a descending sequence of positive integers until reaching 1. This sequence helps count how many different ways objects can be arranged sequentially. It’s a crucial building block for understanding permutations.

Arrangement

In the context of permutations, an arrangement refers to the sequence or order in which a set of objects is organized. When dealing with arrangements, each different sequence that can be made with the object's available is considered unique. For example, with 5 different people, any different sequence of arranging these people counts as a distinct arrangement.

Arrangements are crucial because they consider the order, which can significantly change the outcome. If you switch the positions of two objects, the sequence changes entirely to another arrangement. This is why factorials are used, as they help calculate all possible arrangements without missing any possibilities.

Permutation Formula

Permutations are about finding how many different ways you can arrange a set of objects. The permutation formula is based on factorials and seeks to compute all possible arrangements for a given number of items. For a set of 'n' distinct items, the total number of permutations is calculated using the formula: \( n! \).

The permutation formula is particularly useful in scenarios where you need to count specific sequences, such as seating arrangements, passwords, or order of tasks. This is in contrast to combinations, where the order does not matter. Understanding permutations is key to solving problems that require listing or counting all possible ordered outcomes.