/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Evaluate each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 3

Evaluate each expression. $$ 8 ! $$

Short Answer

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Step by step solution

01

Understand the Factorial Notation

The exclamation mark (!) represents a factorial. The factorial of a non-negative integer n is the product of all positive integers less than or equal to n.
02

Write Down the Expression

Write down the factorial expression for 8!. This means multiplying every whole number from 1 up to 8.
03

Calculate Step-by-Step

First, multiply the initial few numbers: 1 x 2 = 2 2 x 3 = 6 6 x 4 = 24 24 x 5 = 120 120 x 6 = 720 720 x 7 = 5040 5040 x 8 = 40320
04

Write the Final Answer

The result of multiplying all these numbers together is 40320.

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Key Concepts

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

Factorial Notation
Factorial notation is a mathematical concept denoted by the exclamation mark (!). It is used to represent the product of an integer and all the positive integers below it. For example, the expression for the factorial of 8, written as 8!, involves multiplying the numbers from 1 to 8 together. The expression can be written as: 8! = 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1. The factorial of a number n is calculated as: \[ n! = n \times (n-1) \times (n-2) \times \txt{\textellipsis{}} \times 3 \times 2 \times 1 \] Understanding factorial notation is essential in solving various mathematical problems, especially those involving permutations and combinations.
Multiplication
Multiplication is a fundamental arithmetic operation that involves combining equal groups. In the context of factorials, multiplication is used to combine all integers from the number you start with down to 1. Let's take 8! as an example. You multiply the numbers sequentially:
  • 1 脳 2 = 2
  • 2 脳 3 = 6
  • 6 脳 4 = 24
  • 24 脳 5 = 120
  • 120 脳 6 = 720
  • 720 脳 7 = 5040
  • 5040 脳 8 = 40320
This sequence shows step-by-step how to compute the factorial of 8. At each step, you multiply the result of your previous step by the next integer in the sequence until you reach 8.
Integer Products
When dealing with factorials, you work with products of integers. An integer product refers to the result of multiplying whole numbers. For example, calculating 8! involves multiple integer products:
  • First product: 1 脳 2 = 2
  • Second product: 2 脳 3 = 6
  • Third product: 6 脳 4 = 24
As you continue multiplying the integers, the products get progressively larger. Factorials grow very quickly, even for relatively small numbers like 8. The integer products in factorial calculations are crucial in many advanced fields like combinatorics, where they help determine the number of ways a set of items can be arranged.

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