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Chapter 15: Problem 13

Evaluate. $$5 !$$

Short Answer

Expert verified
The factorial of 5 (5!) is calculated by multiplying all the positive integers from 1 to 5: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

Step by step solution

01

Definition of a Factorial

The factorial of a number (n) is the product of all positive integers from 1 to n. It is denoted as n! and can be calculated as \(n! = n\times (n-1) \times...\times 2 \times 1\).
02

Evaluating 5!

Following the definition of the factorial, we will multiply all the positive integers from 1 to 5: $$5! = 5 \times 4 \times 3 \times 2 \times 1$$
03

Simplifying the Expression

Now, let's perform the multiplication: $$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$
04

Final Answer

The factorial of 5 (5!) is equal to 120.

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Key Concepts

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

Definition of Factorial
A factorial, represented by the symbol "!", is an important concept in mathematics. It refers to the operation of multiplying a sequence of descending positive integers down to 1. Specifically, the factorial of a number \( n \), written \( n! \), is the multiplicative sum of all positive integers up to and including \( n \). This can be expressed as \( n! = n \times (n-1) \times (n-2) \times \, ... \, \times 3 \times 2 \times 1 \). This sequence ends with 1 because multiplying by 1 leaves the product unchanged, serving as our stopping point.
Evaluating Factorials
To evaluate a factorial, you simply follow the definition by multiplying the sequence of integers. Taking \( 5! \) as an example, you multiply the numbers starting from 5 and counting down to 1:
  • First, multiply 5 and 4 to get 20.
  • Then, multiply the result, 20, by 3 to get 60.
  • Next, multiply 60 by 2 to get 120.
  • Finally, multiply 120 by 1 (though this doesn't change the value, it completes the factorial process).
Thus, \( 5! = 120 \). Evaluating factorials can involve quite large numbers, especially as \( n \) increases, but the principle remains the same.
Integer Multiplication
Integer multiplication is a basic arithmetic process where two whole numbers (integers) are multiplied to get a product. This operation is central to calculating factorials. When you multiply integers:
  • Ensure each step is executed accurately to avoid errors in your final result.
  • Remember the order doesn't matter (commutative property), hence \( a \times b = b \times a \).
  • Being systematic can save time and reduce mistakes, especially with longer sequences.
In factorials, integer multiplication becomes repetitive, dealing with a series of descending integers. Mastery of integer multiplication is key for ease in evaluating factorials.
Algebra Basics
Understanding algebra basics can greatly aid in working with factorials, especially when simplifying expressions. Algebra involves manipulating symbols and numbers to solve equations or illustrate patterns. In the context of factorials:
  • Algebraic rules can simplify factorial calculations, by allowing for substitution or cancellation when solving equations.
  • For example, in expressions like \( n! = n \times (n-1)! \), knowing that factorial multiplication builds upon itself can help in solving problems methodically.
  • Grasping how factorials relate to algebraic expressions is relevant in more complex mathematics such as permutations or combinations.
Understanding these basics strengthens your ability to process and simplify mathematical challenges in factorial problems.

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