Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-22,\) evaluate each expression. $$ 5 ! $$

Step-by-Step Solution

Answer

The value of \(5!\) is 120.

Step 1: Understanding the Problem

We are asked to evaluate the expression \(5!\). This involves finding the factorial of 5.

Step 2: Understanding Factorial

The factorial of a number \( n \) is represented by \( n! \) and is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).

Step 3: Calculating the Factorial

Calculate \( 5! \) step-by-step:- Calculate \( 5 \times 4 = 20 \).- Multiply the result by the next number: \( 20 \times 3 = 60 \).- Continue with \( 60 \times 2 = 120 \).- Finally, \( 120 \times 1 = 120 \).The factorial of 5 is 120.

Key Concepts

Factorial Calculation

A factorial is a mathematical operation that plays a fundamental role in various branches of mathematics and science. The factorial of a non-negative integer is denoted by the symbol "!" following the number. It represents the product of all positive integers up to that number. For instance, the factorial of 5, which is written as \(5!\), would be calculated by multiplying 5 by every positive integer less than it, down to 1.

Here’s how the factorial operation works:

• Start with the number you want the factorial for, say \( n = 5 \).
• Multiply it by every positive integer less than itself: \( n \times (n-1) \times (n-2) \times \, ... \, \times 2 \times 1 \).
• Stop at 1, since multiplying by any number below that keeps giving you the same number.

The main takeaway is that factorial calculation is straightforward once you understand it involves continuous multiplication until you reach one.

Product of Consecutive Integers

The idea of the product of consecutive integers is central to understanding factorials. When we say consecutive integers, we mean numbers that follow each other in order without any gaps. Factorials are perfect examples of using consecutive integers, as they include multiplying a series of reducing integers from your starting number down to 1.

Let's take a specific factorial as an example:

• For \(5!\), we multiply the sequence: 5, 4, 3, 2, and 1 together.
• Start from 5, then move sequentially downwards: \(5 \times 4 \times 3 \times 2 \times 1\).
• Every step involves multiplying by the next integer in the line until reaching 1.

Each integer plays a role in building up to the total product, which is why understanding the product of consecutive integers makes evaluating factorials logical and methodical.

Mathematical Expressions Evaluation

Evaluating mathematical expressions is a process of decoding and simplifying them to find the value they represent. With factorials, this means taking the initial number and reducing it step-by-step through multiplication. To effectively evaluate a factorial expression like \(5!\), follow a structured calculation sequence:

• Start by writing down the expression: \(5!\) translates to \(5 \times 4 \times 3 \times 2 \times 1\).
• Multiply each pair of numbers sequentially: \(5 \times 4 = 20\).
• Continue multiplying the result by the next integer: \(20 \times 3 = 60\), \(60 \times 2 = 120\).
• Finally, multiply by 1, which leaves you with the same number: \(120 \times 1 = 120\).

By approaching such expressions systematically, you ensure precision and reduce the likelihood of error. Understanding this process is crucial for tackling not just factorials but a wide array of mathematical problems that involve reducing and simplifying expressions.