/*! 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 18 Evaluate each expression. $$ ... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 18

Evaluate each expression. $$ 6 ! \cdot 7 $$

Short Answer

Expert verified
5040

Step by step solution

01

Understand the Factorial Notation

The factorial of a number, denoted by the symbol '!', is the product of all positive integers less than or equal to that number. For example, 6! means to multiply all the numbers from 1 to 6 together.
02

Calculate the Factorial

Calculate 6! by multiplying the integers from 1 to 6: \[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \] \[ 6! = 720 \]
03

Multiply by 7

Now, multiply the result of 6! by 7: \[ 720 \times 7 = 5040 \]

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Key Concepts

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

Factorial Function
The factorial function, denoted by the symbol '!', is a mathematical operation that takes a positive integer and multiplies all the positive integers less than or equal to that number. For example, the factorial of 5 (written as 5!) is calculated as follows: \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]Factorial notation is used very often in permutations, combinations, and other areas of mathematics. It's important to understand that factorials grow very quickly! For instance, while 5! equals 120, 6! equals 720, and 7! equals 5040.
Multiplication
Multiplication is one of the basic arithmetic operations where we add a number to itself a certain number of times. For example, when you multiply 4 by 3, it is the same as adding 4 three times: \[4 \times 3 = 4 + 4 + 4 = 12\]When we use multiplication involving factorials, the values can become very large very quickly because of the nature of factorial growth. In our exercise, we first found 6! which is 720, and then multiplied it by 7 to get 5040. Multiplication helps simplify repeated addition into a more straightforward operation.
Integer Operations
Integer operations involve basic arithmetic operations such as addition, subtraction, multiplication, and division performed with whole numbers. When dealing with factorials, we are primarily utilizing multiplication with integers. For example, in our problem, we are multiplying 6! (an integer value of 720) by another integer, 7. Combining integer operations with the factorial function requires careful computation to avoid mistakes, especially since the numbers grow large quickly. Understanding how to handle these operations is crucial for higher-level mathematics involving sequences, series, and even algorithms.

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Most popular questions from this chapter

A vehicle depreciates by \(20 \%\) of its value each year. If it cost \(\$ 35,000\) new, what is its value after 6 yr?

Determine an expression for the general term \(a_{n}\) of each sequence \(4,8,12,16, \ldots\)

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ 1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \ldots $$

Write the first five terms of each sequence. $$ a_{n}=n-\frac{1}{n} $$

If the given sequence is geometric, find the common ratio \(r .\) If the sequence is not geometric, say so. See Example 1. $$ 5,15,45,135, \ldots $$
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