Factorial Calculator Calculate the factorial of any number (n!) with digit count and expansion.
Factorial Calculator
Calculate the factorial of any number (n!) with digit count and expansion.
Enter a Number
Input a non-negative integer (up to 1000) to calculate its factorial.
View the Result
See n! displayed in full, along with the total number of digits in the result.
Check Expansion
For smaller numbers, view the complete multiplication expansion (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120).
What Is Factorial Calculator?
The factorial of a non-negative integer n, denoted n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1 (the empty product). Factorials grow extremely rapidly — 20! already exceeds 2.4 quintillion, and 100! has 158 digits. Factorials are foundational in combinatorics for counting arrangements (permutations) and selections (combinations), in probability theory, in Taylor series expansions of functions like e^x and sin(x), and in algorithm analysis. This calculator uses BigInt arithmetic to compute exact factorials up to 1000!, producing results with thousands of digits without any rounding or approximation. It also shows the digit count and, for smaller values, the complete multiplication expansion for educational reference.
Why Use Factorial Calculator?
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Computes exact factorials up to 1000! using BigInt precision
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Shows the total digit count of the result
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Displays the multiplication expansion for learning
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No overflow issues — works beyond standard number limits
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Instant calculation even for very large factorials
Common Use Cases
Combinatorics
Calculate permutations and combinations that involve factorial expressions.
Probability
Compute probabilities involving arrangements, such as the birthday problem or lottery odds.
Mathematics Education
Demonstrate how quickly factorials grow and verify homework answers.
Algorithm Analysis
Understand the scale of O(n!) algorithms in computer science.
Technical Guide
The factorial function is defined recursively as n! = n × (n−1)! with the base case 0! = 1. This calculator uses an iterative approach with JavaScript BigInt to avoid stack overflow and maintain exact integer precision. BigInt allows arbitrarily large integers, so 1000! (which has 2,568 digits) is computed exactly without floating-point rounding. The computation is a simple loop: start with result = 1 and multiply by each integer from 2 to n. Time complexity is O(n) multiplications, but each multiplication involves increasingly large numbers, so actual time grows faster in practice. Stirling's approximation (n! ≈ √(2πn) × (n/e)^n) can estimate factorial magnitude: the number of digits in n! is approximately n×log10(n/e) + 0.5×log10(2πn). The calculator caps input at 1000 to maintain reasonable browser performance, though BigInt could theoretically handle larger values.
Tips & Best Practices
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10! = 1 by definition (the empty product convention)
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2Factorials grow incredibly fast — 20! is already over 2 quintillion
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3The number of digits in n! is approximately n×log₁₀(n) − n×log₁₀(e) + 0.5×log₁₀(2πn)
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4Factorials are only defined for non-negative integers; the gamma function generalizes to all numbers
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5For combinations, you can often cancel factorials before multiplying to keep numbers manageable
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🔢 Math & CalculatorsFrequently Asked Questions
Q Why does 0! equal 1?
Q How large can the input be?
Q Are factorials defined for negative numbers?
Q Why do factorials grow so fast?
Q What are factorials used for?
About This Tool
Factorial Calculator is a free online tool by FreeToolkit.ai. All processing happens directly in your browser — your data never leaves your device. No registration or installation required.