Factorial Calculator
Calculate factorials (n!) for any non-negative integer with precision support for very large numbers. Perfect for combinatorics, probability, and mathematical research.
Factorial Calculator
Calculate factorials (n!) for any non-negative integer with support for very large numbers
Factorial (n!): The product of all positive integers from 1 to n. For example: 5! = 5 × 4 × 3 × 2 × 1 = 120
Understanding Factorials
Definition and Notation
The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. By mathematical convention, 0! = 1.
Mathematical Definition:
The factorial function grows extremely rapidly, making it one of the fastest-growing elementary functions in mathematics. Even relatively small values like 70! produce numbers with over 100 digits.
First 10 Factorials
| n | n! | Value |
|---|---|---|
| 0 | 0! | 1 |
| 1 | 1! | 1 |
| 2 | 2! | 2 |
| 3 | 3! | 6 |
| 4 | 4! | 24 |
| 5 | 5! | 120 |
| 6 | 6! | 720 |
| 7 | 7! | 5,040 |
| 8 | 8! | 40,320 |
| 9 | 9! | 362,880 |
| 10 | 10! | 3,628,800 |
Notice how rapidly the values grow. 20! already exceeds 2 quintillion!
Mathematical Applications
Combinatorics and Permutations
Factorials are fundamental to counting problems, particularly in determining the number of ways to arrange objects in different orders.
Permutations
The number of ways to arrange n distinct objects is n!
Arrangements = 5! = 120 ways
Partial Permutations
Arranging r objects from n objects: P(n,r) = n!/(n-r)!
P(8,3) = 8!/5! = 336 ways
Probability and Statistics
Combinations
C(n,r) = n! / (r!(n-r)!)
Used in lottery calculations, poker hands, and statistical sampling
Poisson Distribution
P(X=k) = (λᵏe⁻λ) / k!
Models rare events and arrival processes
Taylor Series
f(x) = Σ(f⁽ⁿ⁾(a)/n!) × (x-a)ⁿ
Fundamental in calculus and function approximation
Calculation Methods and Optimization
Algorithm Implementation
Our calculator uses different approaches based on the size of the input number to ensure optimal performance and accuracy.
Small Numbers (≤20)
Uses standard integer arithmetic with detailed step-by-step breakdown
Step 1: 1! = 1
Step 2: 2! = 2
Step 3: 3! = 6
Step 4: 4! = 24
Step 5: 5! = 120
Large Numbers (>20)
Uses BigInt arithmetic to handle arbitrarily large results with full precision
(158 digits total)
Precision: Exact
Memory: Optimized
Performance Characteristics
| Input Size | Result Digits | Calculation Time | Memory Usage |
|---|---|---|---|
| 1-10 | 1-7 | <1ms | Minimal |
| 11-50 | 8-65 | <5ms | Low |
| 51-100 | 66-158 | <10ms | Moderate |
| 101-1000 | 159-2568 | 10-100ms | Higher |
How to Use the Factorial Calculator
Step-by-Step Guide
Enter Your Number
Input any non-negative integer from 0 to 1000. The calculator will show a preview of what will be calculated.
Calculate Factorial
Click "Calculate Factorial" to compute the result. The tool automatically selects the optimal algorithm for your input size.
Explore the Result
View the complete result with digit count, calculation time, and mathematical properties. Large numbers are displayed with smart formatting.
View Calculation Steps
Enable "Show Steps" to see the detailed calculation process, perfect for educational purposes and verification.
Practical Examples
Combinatorics Problem
How many ways can 8 people sit around a circular table?
= (8-1)! = 7!
= 5,040 ways
Probability Calculation
What's the probability of getting exactly 3 heads in 10 coin flips?
= 3,628,800/(6×5,040)
= 120 ways
Large Number Analysis
Calculate 50! to understand the scale of very large factorials.
65 digits long
Calculation: <5ms
Educational Demonstration
Show students how factorials grow by comparing small values.
6! = 720 (6× larger)
Growth factor = 6
Mathematical Properties and Insights
Interesting Factorial Properties
Growth Rate
Factorials grow faster than exponential functions but slower than double exponentials.
- • n! grows approximately as √(2πn) × (n/e)ⁿ (Stirling's approximation)
- • 70! ≈ 1.2 × 10¹⁰², exceeding a googol
- • Each step multiplies the previous result by n
Trailing Zeros
The number of trailing zeros in n! equals the number of times 10 divides n!.
- • Count factors of 5: ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + ...
- • 25! has 6 trailing zeros
- • 100! has 24 trailing zeros
Special Values and Identities
Base Cases
- 0! = 1 (empty product)
- 1! = 1 (identity)
- (-1)! = undefined
Useful Identities
- n! = n × (n-1)!
- (2n)! = 2²ⁿ × n! × (2n-1)!!
- n! ≈ √(2πn) × (n/e)ⁿ
Real-World Context
Astronomical Scale
70! ≈ 1.2 × 10¹⁰², which far exceeds the estimated number of atoms in the observable universe (≈10⁸⁰).
Practical Limits
Most real-world combinatorial problems involve factorials well under 100!, making our calculator suitable for virtually all applications.
Computing Context
Standard 64-bit integers overflow at 21!, highlighting the importance of BigInt arithmetic for larger calculations.
Frequently Asked Questions
Why is 0! equal to 1?
This is defined by mathematical convention and makes many formulas work correctly. It represents the number of ways to arrange zero objects, which is exactly one way (doing nothing). This definition ensures continuity with the recursive formula n! = n × (n-1)!.
What's the largest factorial I can calculate?
Our calculator supports factorials up to 1000!, though very large values may take longer to compute. 1000! has over 2,500 digits! For most practical applications, factorials larger than 100! are rarely needed.
How accurate are the large number calculations?
Our calculator uses BigInt arithmetic, which provides exact precision for all calculations. Unlike floating-point approximations, every digit in the result is mathematically correct, no matter how large the number becomes.
Can I calculate negative factorials?
Factorials are only defined for non-negative integers. For negative numbers, mathematicians use the gamma function: Γ(n) = (n-1)!, but this involves complex analysis and is beyond the scope of basic factorial calculation.
What's the difference between n! and n^n?
n! multiplies all integers from 1 to n, while n^n multiplies n by itself n times. For example: 5! = 5×4×3×2×1 = 120, but 5^5 = 5×5×5×5×5 = 3,125. Generally, n^n grows faster than n!.
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