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

Number (n)

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:

n! = n × (n-1) × (n-2) × ... × 2 × 1

0! = 1 (by definition)

1! = 1

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!

Example: 5 books on a shelf
Arrangements = 5! = 120 ways

Partial Permutations

Arranging r objects from n objects: P(n,r) = n!/(n-r)!

Example: Choose 3 from 8
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

5! = 5 × 4 × 3 × 2 × 1
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

100! = 93326215443944152681...
(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?

Circular arrangements = (n-1)!
= (8-1)! = 7!
= 5,040 ways

Probability Calculation

What's the probability of getting exactly 3 heads in 10 coin flips?

C(10,3) = 10!/(3!×7!)
= 3,628,800/(6×5,040)
= 120 ways

Large Number Analysis

Calculate 50! to understand the scale of very large factorials.

50! = 3.04×10⁶⁴
65 digits long
Calculation: <5ms

Educational Demonstration

Show students how factorials grow by comparing small values.

5! = 120
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!.