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Chapter 11: Problem 11

Evaluate. $$(8-3) !$$

Short Answer

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Step by step solution

01

Understand the Problem

The expression asks to evaluate \( (8-3)! \). First, determine what is inside the parentheses.
02

Calculate the Value Inside the Parentheses

Subtract 3 from 8: \( 8 - 3 = 5 \). So, \( (8 - 3) = 5 \).
03

Evaluate the Factorial

Now, find the factorial of 5, which is denoted as \( 5! \). The factorial of a number \( n \) is the product of all positive integers from 1 to \( n \). Hence, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).
04

Multiply the Numbers

Calculate the product step-by-step: \( 5 \times 4 = 20 \, 20 \times 3 = 60 \, 60 \times 2 = 120 \, 120 \times 1 = 120 \). Therefore, \( 5! = 120 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorial
A factorial, represented by an exclamation mark (!), is a mathematical operation that multiplies a sequence of descending positive integers. To find the factorial of a number \( n \), written as \( n! \), you multiply all whole numbers from \( n \) down to 1. For example, if \( n \) is 5, then the factorial is calculated as: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] Factorials are useful in permutations and combinations, which are topics in probability and statistics. Understanding how to calculate a factorial is essential for solving more complex mathematical problems, such as those found in algebra and calculus.
Order of Operations
Order of operations is a set of rules used to determine the order in which different operations in a mathematical expression are carried out. It ensures that everyone solves the problem the same way to get a consistent result. The common acronym used to remember the order is PEMDAS:
  • **P**arentheses first
  • **E**xponents (i.e., powers and square roots, etc.)
  • **M**ultiplication and **D**ivision (left-to-right)
  • **A**ddition and **S**ubtraction (left-to-right)
In the given exercise, we first needed to solve the operation inside the parentheses: \[ 8 - 3 \] After solving this, we then proceeded to evaluate the factorial as instructed by the rules of operations.
Basic Arithmetic
Basic arithmetic forms the foundation of all mathematical concepts. It includes four primary operations: addition, subtraction, multiplication, and division. In the exercise, subtraction was the first arithmetic operation performed. We calculated \[ 8 - 3 = 5 \] This simplified our problem to finding the factorial of 5, which then required multiplication:
  • Multiply 5 by 4 to get 20
  • Multiply 20 by 3 to get 60
  • Multiply 60 by 2 to get 120
  • Multiply 120 by 1 to confirm 120
Understanding these basic operations is crucial for more complex mathematical topics, so practice regularly to ensure a solid grasp of these concepts.

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