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Chapter 6: Problem 23

Evaluate the factorial expression. $$7 !-6 !$$

Short Answer

Expert verified
The result of the factorial expression 7! - 6! is 4320.

Step by step solution

01

Compute Factorial of 7

Calculate the factorial of 7 using the property of factorials, i.e., 7! = 7 * 6!. Given 6! = 720 (as 6*5*4*3*2*1 = 720), we have 7! = 7 * 720 = 5040.
02

Subtract Factorial of 6 from Factorial of 7

The next step is to subtract 6! from the result in step 1. We already know that 6! equals 720. So, perform the subtraction 5040 - 720 = 4320.

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Key Concepts

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

factorial calculation
Factorial calculation is a fundamental concept, especially in the field of mathematics. The factorial of a non-negative integer, denoted by the symbol \(!\), is the product of all positive integers less than or equal to that number. For example, to calculate \(7!\), we multiply all whole numbers from 1 to 7:
  • 7! = 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1
This results in \( 5040 \). Factorials are used in various mathematical calculations, ranging from permutations and combinations to more advanced calculus operations. Understanding how to calculate factorials forms the basis of more complex mathematical problem-solving.
arithmetic operations
Arithmetic operations are the building blocks of mathematics, consisting of addition, subtraction, multiplication, and division. In this exercise, subtraction is used to compare two factorial expressions:
  • From step 1, the calculated value of \(7!\) is \(5040\).
  • From step 2, we subtract the value of \(6!\), which is \(720\), from \(5040\).
This operation results in \(4320\) (i.e., \(5040 - 720 \)). Arithmetic operations allow us to manipulate numbers in a variety of ways to solve equations and problems. Recognizing the correct operation is crucial for arriving at the accurate solution.
precalculus problem-solving
Precalculus problem-solving often involves the application of foundational mathematical concepts to solve more complex problems. Problems like the one in this exercise are typical precalculus types, where understanding factorials and arithmetic operations is essential. To tackle these types of problems efficiently:
  • First, break down the problem into smaller parts. For instance, compute each factorial separately as step 1 illustrates with \(7!\).
  • Next, apply arithmetic operations, such as the subtraction demonstrated in step 2.
  • Finally, validate your results with logical reasoning or re-calculation to ensure accuracy.
Precalculus lays the groundwork for calculus by nurturing skills in algebra, geometry, and trigonometry, helping students to build confidence in problem-solving and critical thinking.

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Most popular questions from this chapter

The graph of $$r=1.5^{\sin \theta}-2.5 \cos 4 \theta+\sin ^{7} \frac{\theta}{15}$$ is a butterfly curve similar to the one shown below. (GRAPH CAN'T COPY) Use a graphing utility to graph the butterfly curve for a. \(0 \leq \theta \leq 5 \pi\) b. \(0 \leq \theta \leq 20 \pi\) For additional information on butterfly curves, read "The Butterfly Curve" by Temple H. Fay, The American Mathematical Monthly, vol. \(96,\) no. 5 (May 1989 ), \(\mathrm{p} .442\)

Approximate \((1.02)^{8}\) by evaluating the first three terms of \((1+0.02)^{s}\)

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