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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: \(1+99=100 .)\)

Match the expression with the most appropriate expression from the column on the right. ____ \(\sum_{k=1}^{4} k^{2}\) a) \(-1+1+(-1)+1\) b) \(a_{2}=25\) c) \(a_{2}=8\) d) \(\sum_{k=1}^{4} 5 k\) e) \(S_{3}\) f) \(1+4+9+16\)

Form 1 of the binomial theorem can be proved using form 2 of the binomial theorem. The key step in that proof is showing that the coefficients inside Pascal's triangle are found by adding the two terms above. Prove this fact by showing that $$\left(\begin{array}{l}{n} \\\\{r}\end{array}\right)=\left(\begin{array}{c}{n-1} \\ {r-1}\end{array}\right)+\left(\begin{array}{c}{n-1} \\\\{r}\end{array}\right)$$

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

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