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Chapter 11: Problem 12

Simplify. $$9 !$$

Short Answer

Expert verified
9! = 362880

Step by step solution

01

Understand the notation

Recognize that the exclamation mark '!' denotes a factorial. The factorial of a number n, denoted as n!, is the product of all positive integers less than or equal to n.
02

Write out the factorial

Write 9! as a product of integers from 9 down to 1: 9! = 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1
03

Multiply the factors

Calculate the product step by step: 9 脳 8 = 72 72 脳 7 = 504 504 脳 6 = 3024 3024 脳 5 = 15120 15120 脳 4 = 60480 60480 脳 3 = 181440 181440 脳 2 = 362880 362880 脳 1 = 362880

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Key Concepts

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

Notation
In mathematics, the exclamation mark '!' is called a factorial. When you see it after a number, it means you need to multiply that number by all the positive integers less than it.
For example, the notation for the factorial of 9 is written as 9!. It tells us to find the product of all numbers from 9 down to 1.
So, 9! = 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1.
  • Factorials are commonly used in permutations, combinations, and other areas of mathematics.
  • Knowing factorial notation helps you understand how to organize and count things.
Product of Integers
The core idea behind factorials is multiplying a series of integers. For 9!, we multiply all whole numbers from 9 down to 1.
Here is how it breaks down:
  • 9 脳 8 = 72
  • 72 脳 7 = 504
  • 504 脳 6 = 3024
  • 3024 脳 5 = 15120
  • 15120 脳 4 = 60480
  • 60480 脳 3 = 181440
  • 181440 脳 2 = 362880
  • 362880 脳 1 = 362880

Multiplying these numbers is the essence of finding the product of integers in a factorial.
It shows you how numbers combine together step-by-step to form a large product. Understanding this helps you grasp the magnitude of factorials quickly.
Step-by-step Multiplication
Breaking down the multiplication process into small steps makes it easier to manage.
Instead of multiplying all numbers at once, you multiply two numbers at a time.
Here is a simple way to do that:
  • First, start with the top two numbers: 9 脳 8 = 72.
  • Then, use the result and multiply by the next number: 72 脳 7 = 504.
  • Continue the process: 504 脳 6 = 3024, 3024 脳 5 = 15120, and so on.

Each step builds on the previous one. This method helps you keep track of the calculations without getting overwhelmed.
  • This approach also reduces the chances of making mistakes in longer multiplication sequences.
  • Step-by-step multiplication is a useful technique in many areas of math, not just factorials.

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Most popular questions from this chapter

It is said that as a young child, the mathematician Karl F. Gauss \((1777-1855)\) was able to compute the sum \(1+2+3+\cdots+100\) very quickly in his head. Explain how Gauss might have done this and present a formula for the sum of the first \(n\) natural numbers. (Hint: \(1+99=100 .)\)

Write out and evaluate each sum. $$ \sum_{k=0}^{5}\left(k^{2}-2 k+3\right) $$

Find the common ratio for each geometric sequence. $$75,15,3, \frac{3}{5}, \dots$$

Solve. $$|x-3|=11$$

Find the first term and the common difference. Find \(a_{1}\) and \(d\) if \(a_{12}=24\) and \(a_{25}=50\)
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