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Chapter 12: Problem 4

Evaluate each expression. $$ 9 ! $$

Short Answer

Expert verified
The value of 9! is 362880.

Step by step solution

01

Understand Factorial Notation

The factorial of a non-negative integer is the product of all positive integers less than or equal to that number. It is denoted by following the number with an exclamation point (!). For example, the factorial of 4, written as 4!, is calculated as 4 脳 3 脳 2 脳 1.
02

Set Up the Factorial

To evaluate 9!, write out the expression as the product of all positive 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

Perform the Multiplication

Multiply the integers together 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.
04

Write the Final Answer

After performing the multiplications, the final value of 9! is 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 mathematical concept represented by an exclamation point (!) following a non-negative integer. It signifies the product of all positive integers less than or equal to that number. For instance, with 4!, you multiply 4 by all the integers below it: 4 脳 3 脳 2 脳 1.
This notation is crucial in various areas of mathematics, including combinations, permutations, and other probability-related problems.
Multiplication Steps
When evaluating a factorial, you follow specific multiplication steps to find the product of the sequence of numbers. For the factorial of 9 (written as 9!), you set it up by listing all integers from 9 down to 1. To get: 9! = 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1.
Here's how you can perform the multiplication 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
Following these steps ensures you correctly calculate the product.
Integer Products
The process of determining a factorial involves calculating the product of integers. Each multiplication step results in an integer product, which is then used in the next multiplication until the final product is reached.
In the case of 9!, the integer products formed at each step are:
  • Initial multiplication: 9 脳 8 = 72
  • Subsequent steps build on this result: 72 脳 7 = 504, and so on
This sequence demonstrates the importance of understanding how multiplication of large numbers operates step-by-step.

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Most popular questions from this chapter

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