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Chapter 5: Problem 8

Find the value of each factorial. \(12!\)

Short Answer

Expert verified
479001600

Step by step solution

01

Understand the definition of a factorial

A factorial, denoted by an exclamation mark (!), is the product of all positive integers from 1 up to that number. For example, \( n! = n \times (n-1) \times (n-2) \times \text{...} \times 1 \).
02

Apply the factorial definition to 12!

Compute the product of all positive integers from 1 to 12: \( 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).
03

Multiply step-by-step

First compute smaller groups and then multiply those results together:\(12 \times 11 = 132\)\(132 \times 10 = 1320\)\(1320 \times 9 = 11880\)\(11880 \times 8 = 95040\)\(95040 \times 7 = 665280\)\(665280 \times 6 = 3991680\)\(3991680 \times 5 = 19958400\)\(19958400 \times 4 = 79833600\)\(79833600 \times 3 = 239500800\)\(239500800 \times 2 = 479001600\)\(479001600 \times 1 = 479001600\)

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Key Concepts

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

Factorial Computation
A factorial is a way to multiply a series of descending natural numbers. It's denoted using an exclamation mark, like this: !. The basic idea is to take a number and multiply it by all the numbers below it down to 1. For example, 5! means 5 x 4 x 3 x 2 x 1. This is very useful in mathematical problems involving permutations and combinations.
In this exercise, we are asked to find 12!. This means we multiply 12 by every whole number less than 12 down to 1. The expression looks like this: \[12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\]
Mathematical Operations
To solve the problem, you need to perform several multiplication operations. It might seem complicated, but taking it step by step makes it manageable. The key here is understanding multiplication and how to handle large numbers.
When dealing with large multiplications, you can break them down into smaller steps.
  • Step 1: Multiply 12 by 11 to get 132.
  • Step 2: Multiply the result by 10 to get 1320.
  • Step 3: Continue this process until you've included all numbers down to 1.
This chunking method simplifies what could be a daunting calculation.
Step-by-Step Solutions
Breaking down a problem into smaller steps helps you stay organized and ensures accuracy. Here's how to calculate 12! step by step:
  • 12 x 11 = 132
  • 132 x 10 = 1320
  • 1320 x 9 = 11880
  • 11880 x 8 = 95040
  • 95040 x 7 = 665280
  • 665280 x 6 = 3991680
  • 3991680 x 5 = 19958400
  • 19958400 x 4 = 79833600
  • 79833600 x 3 = 239500800
  • 239500800 x 2 = 479001600
  • 479001600 x 1 = 479001600
Each step involves multiplying the result from the previous step by the next number in the sequence. By following these steps, you ensure that you correctly get the final result, 479001600.

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