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Chapter 1: Problem 65

Evaluate. \(5 !\).

Short Answer

Expert verified
The solution to \(5 !\) is 120.

Step by step solution

01

Understanding Factorial

The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. It is formulaically represented as \(n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1\)
02

Calculate \(5 !\)

To calculate \(5 !\), multiply 5 by each positive integer less than it all the way down to 1. This results in \(5 ! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

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Key Concepts

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

Integer Multiplication
Multiplication is one of the fundamental arithmetic operations and is crucial in understanding the concept of factorials. It involves the process of adding a number to itself a certain number of times. For example, when we say 3 \( \times \) 2, it means adding 3 twice: 3 + 3, which equals 6.
In the concept of factorials, integer multiplication is used repeatedly.
  • An integer is a whole number that can be positive, negative, or zero.
  • In a factorial, we deal specifically with non-negative integers.

This means that when calculating a factorial, we multiply together a series of consecutive whole numbers, starting from the number in question (like 5) down to 1. Each multiplication builds on the previous result, leading us to the final product.
Mathematical Notation
Mathematical notation helps communicate complex ideas in a streamlined and universally understood manner. When you see a factorial, it is expressed in a standard way: \( n! \). The exclamation mark (!) signifies a factorial operation.
Here鈥檚 how to break it down using \( 5! \):
  • The number 5 is the integer for which we are finding the factorial.
  • The exclamation mark indicates that a series of specific multiplications are to be performed.

This notation allows mathematicians and students to quickly convey the need to multiply all whole numbers descending from a given integer down to 1. Understanding this notation is vital because it provides a clear instruction of what arithmetic operations need to be performed next.
Sequential Multiplication
Sequential multiplication comes into play when calculating factorials. It involves performing multiplication operations in a specific sequence until all integers in the series have been used. Imagine calculating for \( 5! \), where the sequence of multiplications must be followed accurately.
  • First, you multiply 5 by 4, resulting in 20.
  • Then, take the product (20) and multiply by 3, resulting in 60.
  • Next, multiply 60 by 2, leading to 120.
  • Finally, 120 is multiplied by 1, which keeps the product at 120.

This method ensures all integers are used in the multiplication, and the calculations are performed systematically from the largest integer to the smallest, providing clarity and accuracy in solving factorials.

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Most popular questions from this chapter

Find \(f \circ g\) and \(g \circ f\). $$f(x)=1-x^{2}, g(x)=\sin x$$

Prove that \(\sqrt{3}\) is irrational.

(a) Use a graphing utility to graph \(f_{A}(x)=A \cos x\) for several values of \(A ;\) use both positive and negative values. Compare your graphs with the graph of \(f(x)=\cos x\). (b) Now graph \(f_{B}(x)=\cos B x\) for several values of \(B\). since the cosine function is even, it is sufficient to use only positive values for \(B\). Use some values between 0 and 1 and some values greater than \(1 .\) Again, compare your graphs with the graph of \(f(x)=\cos x\). (c) Describe the effects that the coefficients \(A\) and \(B\) have on the graph of the cosine function.

Give the domain and range of the function. $$f(x)=3 x-2$$

Sketch the graph of the function. $$g(x)=-2 \cos x$$.
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