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Chapter 13: Problem 12

Evaluate each expression. $$ 9 ! $$

Short Answer

Expert verified
9! = 362880.

Step by step solution

01

Understand Factorial Notation

Factorial notation, denoted by an exclamation mark (!), means to multiply a series of descending natural numbers. For instance, if you have number n, factorial (n!) would be n 脳 (n-1) 脳 (n-2)... 脳 1.
02

Write the Expression for 9!

According to the definition of factorial, write down the expression for 9 factorial: 9! = 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1.
03

Perform the Multiplications Step-by-Step

Start multiplying the numbers step-by-step: 1. 9 脳 8 = 72 2. 72 脳 7 = 504 3. 504 脳 6 = 3024 4. 3024 脳 5 = 15120 5. 15120 脳 4 = 60480 6. 60480 脳 3 = 181440 7. 181440 脳 2 = 362880 8. 362880 脳 1 = 362880

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Key Concepts

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

Factorial Notation
Factorial notation is a symbol used in mathematics that describes the product of an integer and all the integers below it. It is denoted by an exclamation mark (!). For example, the factorial of 5 is written as 5!. To understand this better, let's see how you calculate 5!:
  • Start with 5.
  • Multiply it by every positive whole number less than 5: 5 脳 4 脳 3 脳 2 脳 1.
  • The result is 120.

This notation is helpful in various mathematical computations, particularly those involving permutations and combinations.
Multiplication
Multiplication in the context of factorials involves sequentially multiplying descending numbers. For example, if you're calculating 9!, you start at 9 and multiply it by each integer down to 1 (9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1). Let's break this down step-by-step:
  • First, multiply 9 by 8 to get 72.
  • Next, take that result (72) and multiply it by 7 to get 504.
  • Then, multiply 504 by 6, resulting in 3024.
  • Continue the process: 3024 脳 5 = 15120.
  • 15120 脳 4 = 60480.
  • 60480 脳 3 = 181440.
  • 181440 脳 2 = 362880.
  • Finally, 362880 脳 1 = 362880.

Each step is a multiplication operation, emphasizing how factorial calculations grow rapidly in size.
Factorial of a Number
The factorial of a number n (denoted as n!) is a fundamental concept in mathematics. It signifies the product of all positive integers up to n. For example, the factorial of 9 (9!) can be illustrated as:
  • 9! = 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1
  • Calculating each step: 9 脳 8 = 72, then 72 脳 7 = 504, and so on.
  • Ending with 362880.

Factorials are typically used in mathematical fields dealing with combinatorics, such as permutations, where you need to find out how many ways you can arrange a set of objects.

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