Factorial of Hundred (100!) – Definition, Value & Solved Examples

The Factorial of Hundred (100!) is one of the most intriguing concepts in mathematics, representing the product of all positive integers from 1 to 100. While factorials are commonly used in smaller numbers like 5! or 10!, understanding 100! gives us a glimpse into how quickly numbers grow and why factorials are essential in fields like combinatorics, probability, algebra, and computer science.

With its 158 digits and 24 trailing zeros, the Factorial of Hundred (100!) is a powerful example of large numbers that are difficult to compute manually but can be managed efficiently using scientific notation or programming tools. This article will explore the definition of factorial, the exact and approximate value of 100!, solved examples, and practical applications, making it easier for readers of all levels to understand and apply this important mathematical concept.

What is the Factorial of Hundred (100!)

The Factorial of Hundred (100!) is one of the most fascinating concepts in mathematics. At its core, a factorial represents the product of all positive whole numbers from 1 up to a given number. Specifically, 100! is the product of all numbers from 1 to 100:

100!=100×99×98×⋯×3×2×1100! = 100 \times 99 \times 98 \times \dots \times 3 \times 2 \times 1100!=100×99×98×⋯×3×2×1

This enormous number has 158 digits and ends with 24 zeros. Understanding 100! not only gives us a clear perspective on how rapidly factorials grow but also shows the importance of factorials in mathematics, probability, combinatorics, and computer science.

In this article, we will explore the definition of factorial, its exact and approximate values, examples, properties, and applications, all explained in an easy-to-understand language for every type of reader.

What Is the Factorial of a Number?

The factorial of a number, denoted as n!, is the multiplication of all positive integers from 1 up to n. It is one of the most widely used concepts in mathematics.

Mathematical Definition:

n!=n×(n−1)×(n−2)×⋯×3×2×1n! = n \times (n - 1) \times (n - 2) \times \dots \times 3 \times 2 \times 1n!=n×(n−1)×(n−2)×⋯×3×2×1

Examples of Factorials:

• 3!=3×2×1=63! = 3 \times 2 \times 1 = 63!=3×2×1=6
  

• 5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 1205!=5×4×3×2×1=120
  

• 0!=10! = 10!=1 (special case, by definition)
  

Factorials are essential in fields like permutations, combinations, algebra, mathematical analysis, and probability theory.

The Factorial of Hundred (100!) is simply the extension of this concept to the number 100. It is calculated in the same way as smaller factorials but results in a number that is practically impossible to write fully without scientific notation.

Exact and Approximate Value of Factorial of Hundred (100!)

The Factorial of Hundred (100!) produces a number so large that it cannot be easily written or computed manually. To make it understandable, mathematicians use exact value and approximate value.

Exact Value of 100!

The exact value of 100! is:

93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000

• This number has 158 digits.
  

• It ends with 24 trailing zeros, caused by multiple factors of 10 in the product.
  

Approximate Value Using Scientific Notation

Because the exact number is too large, we often use scientific notation to represent it.

100!≈9.332621544×10157100! \approx 9.332621544 \times 10^{157}100!≈9.332621544×10157

• Here, E in calculators (like 9.332621544E+157) represents “×10 to the power of.”
  

• Scientific notation helps simplify very large numbers and makes it easier to read and use in calculations.
  

Understanding “E” in Scientific Notation

The “E” is often seen when writing Factorial of Hundred (100!) in scientific calculators. It simply means “times ten raised to the power of”.

• Example: 9.332621544E+157 is the same as 9.332621544 × 10¹⁵⁷.
  

• This notation is not Euler’s constant (e), but a way to handle large numbers efficiently.
  

Factorial of Hundred Formula

The Factorial of Hundred (100!) can be represented using the standard factorial formula:

General Formula:

n!=n×(n−1)×(n−2)×⋯×2×1n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1n!=n×(n−1)×(n−2)×⋯×2×1

For 100:

100!=100×99×98×⋯×3×2×1100! = 100 \times 99 \times 98 \times \dots \times 3 \times 2 \times 1100!=100×99×98×⋯×3×2×1

• Start multiplying from 100 and continue down to 1.
  

• This multiplication results in an extremely large number, which is why we often use scientific notation.
  

Recursive Formula:

n!=n×(n−1)!n! = n \times (n-1)!n!=n×(n−1)!

• For example:
  

• 8!=8×7!8! = 8 \times 7!8!=8×7!
  

• 9!=9×8!9! = 9 \times 8!9!=9×8!
  

• Using this recursive formula makes it easier to calculate factorials in programming or mathematical software.
  

Solved Examples of Factorial

1. Factorial of 5 (5!)

5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 1205!=5×4×3×2×1=120

Or using recursion:

5!=5×4!=5×24=1205! = 5 \times 4! = 5 \times 24 = 1205!=5×4!=5×24=120

2. Factorial of 10 (10!)

10!=10×9×8×7×6×5×4×3×2×1=3,628,80010! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,80010!=10×9×8×7×6×5×4×3×2×1=3,628,800

3. Factorial of 0 (0!)

By definition,

0!=10! = 10!=1

4. Factorial of Hundred (100!)

100!=100×99×98×⋯×2×1100! = 100 \times 99 \times 98 \times \dots \times 2 \times 1100!=100×99×98×⋯×2×1

• Approximation: 100!≈9.332621544×10157100! \approx 9.332621544 \times 10^{157}100!≈9.332621544×10157
  
  

• Exact Value: 158-digit number ending with 24 zeros.
  

Why Factorials Are Not Defined for Negative Numbers

The factorial function is only defined for non-negative integers.

• Attempting to calculate (−1)!(-1)!(−1)! or (−5)!(-5)!(−5)! leads to undefined results.
  

• This is because factorial relies on the recursive relation n!=n×(n−1)!n! = n \times (n-1)!n!=n×(n−1)!, and applying it to negative integers results in invalid computations.
  

Key Point: Factorials are only valid for 0 and positive integers.

Applications of Factorial of Hundred (100!)

The Factorial of Hundred (100!) is not just a theoretical number; it has practical importance in multiple fields:

1. Combinatorics and Permutations

• Factorials are used to calculate the number of ways to arrange objects.
  

• Example: Number of ways to arrange 100 books on a shelf = 100!.
  

2. Probability and Statistics

• Factorials appear in formulas for permutations and combinations:
  

nCr=n!r!(n−r)!nCr = \frac{n!}{r!(n-r)!}nCr=r!(n−r)!n!

• They are essential in calculating probabilities for events.
  

3. Mathematical Analysis

• Factorials are part of series expansions like Taylor or Maclaurin series.
  

• They help determine the value of functions with infinite sums.
  

4. Computer Science

• Factorials are used in algorithm design, recursion problems, and combinatorial computations.
  

5. Real-World Applications

• Factorials help in data analysis, arrangement problems, and even cryptography.
  

History of Factorial

• Factorials were first used in the 13th century to count permutations.
  

• The modern notation n!n!n! was introduced in the early 19th century by Christian Kramp, a French mathematician.
  

• Since then, factorials have become an integral part of mathematics.
  

What is the Factorial of 100 Voice Command (Speak)

1. Google Assistant / Android

• Say:
  “Hey Google, what is the factorial of 100?”
  
  

• Google Assistant will typically respond with the approximate value in scientific notation, e.g., 9.332621544 × 10^157.
  

2. Siri / Apple Devices

• Say:
  “Hey Siri, calculate 100 factorial”
  

• Siri will return the value in scientific notation or redirect you to a calculator for large numbers.
  

3. Alexa / Amazon Echo

• Say:
  “Alexa, what is 100 factorial?”
  
  

• Alexa may provide the approximate value or suggest using a calculator app because 100! is extremely large.
  

4. Using Calculator Apps with Voice Input

• Open your calculator app on a smartphone or computer.
  

• Use the voice input button and say:
  “100 factorial”
  

• The app will display the exact value or scientific approximation depending on its capability.
  

The Factorial of Hundred (100!) is one of the largest numbers frequently discussed in mathematics. While its exact value has 158 digits, its approximate value 9.332621544×101579.332621544 × 10¹⁵⁷9.332621544×10157 is often used for practical purposes. Factorials are central to algebra, combinatorics, probability, and computer science, and understanding them strengthens your foundation in mathematics.