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MonoCalc

Factorial Calculator

Enter a non-negative integer

Enter any non-negative integer to calculate its factorial

Result

Step-by-Step Calculation:

5! = 5 × 4 × 3 × 2 × 1

Factorial Value:

5! = 120

About the tool

🧮 Factorial Calculator – Calculate n! with Step-by-Step Explanation

Need to calculate a factorial? The Factorial Calculator computes the factorial of any non-negative integer and shows you the step-by-step multiplication process.

This guide explains what factorials are, how they're calculated, and walks you through using our free online factorial calculator to solve factorial problems.

📘 What Is a Factorial?

A factorial is the product of all positive integers less than or equal to a given positive integer. It is denoted by the symbol "!" after the number.

For example, 5! (read as "five factorial") is calculated as:

5! = 5 × 4 × 3 × 2 × 1 = 120

By mathematical definition, 0! equals 1.

⚙️ How the Factorial Calculator Works

Our factorial calculator uses a simple but powerful algorithm to calculate factorials:

  1. It takes your input number (n)
  2. Multiplies all integers from n down to 1
  3. Shows each step of the multiplication process
  4. Handles special cases like 0! and 1!
  5. Uses BigInt for large numbers when needed

🧩 Key Features

  • ⚡ Instant calculations for any non-negative integer
  • 📊 Step-by-step visualization of the multiplication process
  • 🔢 Support for large numbers using BigInt
  • 🔐 Client-side only — no data is ever uploaded
  • 📱 Mobile and desktop-friendly interface

💡 Practical Uses of Factorials

  • 📊 Combinatorics: Calculating permutations and combinations
  • 📈 Probability: Computing probabilities in various distributions
  • 🧪 Statistics: Used in many statistical formulas
  • 🔬 Mathematics: Taylor series expansions, binomial coefficients
  • 💻 Computer Science: Algorithm analysis and recursive functions

For example, to calculate the number of ways to arrange n distinct objects, you use n!

✅ Understanding Factorial Growth

Factorials grow extremely quickly:

  • 5! = 120
  • 10! = 3,628,800
  • 15! = 1,307,674,368,000
  • 20! = 2,432,902,008,176,640,000

This rapid growth makes factorials important in complexity theory and algorithm analysis.

🌟 Examples of Factorial Calculations

Here are some examples of factorial calculations:

  • 0! = 1 (by definition)
  • 1! = 1
  • 4! = 4 × 3 × 2 × 1 = 24
  • 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

🔄 Related Mathematical Concepts

Factorials are connected to several important mathematical concepts:

  • Permutations: P(n,r) = n! / (n-r)!
  • Combinations: C(n,r) = n! / (r! × (n-r)!)
  • Binomial Theorem: Uses factorials in the calculation of binomial coefficients
  • Gamma Function: Extends the factorial to non-integer values
  • Stirling's Approximation: Approximates factorials for large numbers

🔄 How to Use the Factorial Calculator

  1. Enter a non-negative integer in the input field
  2. The calculator will instantly compute the factorial
  3. View the step-by-step calculation process
  4. For 0 and 1, the calculator will show the special case explanation
  5. For large numbers, the result will use BigInt to ensure accuracy

Frequently Asked Questions

  • Is the Factorial Calculator free ?

    Yes, Factorial Calculator is totally free :)

  • Can i use the Factorial Calculator offline ?

    Yes, you can install the webapp as PWA.

  • Is it safe to use Factorial Calculator ?

    Yes, any data related to Factorial Calculator only stored in your browser(if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

  • What is a factorial?

    A factorial is the product of all positive integers less than or equal to a given positive integer. It is denoted by the symbol '!' after the number. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.

  • Why is 0! equal to 1?

    In mathematics, 0! is defined as 1 by convention. This definition makes many mathematical formulas and identities work consistently. It can be understood through the concept of combinations: there is exactly 1 way to arrange 0 objects.

  • What are factorials used for?

    Factorials are used in many areas of mathematics, particularly in combinatorics and probability. They help calculate permutations (ways to arrange items), combinations (ways to select items), and appear in many mathematical formulas including the Taylor series expansion of e^x and in the calculation of binomial coefficients.

  • How large can factorials get?

    Factorials grow extremely quickly. For example, 10! is already 3,628,800, and 20! is about 2.4 quintillion (2.4×10^18). This rapid growth means that for even moderately large numbers, factorials become too large for standard number formats, which is why our calculator uses BigInt for large inputs.