Factorial Calculator
Calculate factorial n! for non-negative integers using BigInt precision, count trailing zeros, find total digit length, and view Stirling's approximation.
n! Factorial Solver
Enter a non-negative integer n (0 ≤ n ≤ 1000).
Factorial Output
Quick Summary
Our factorial calculator n exclamation steps tool evaluates the product of all positive integers less than or equal to $n$ ($n! = n \times (n-1) \dots \times 1$). It provides exact BigInt precision, trailing zeros count, total digit length, and Stirling's asymptotic approximation.
How It Works: Factorial Multiplication & Properties
The factorial operation grows extremely rapidly (super-exponentially):
1. **Definition ($n!$):** $n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1$.
2. **Zero Factorial Convention ($0!$):** By mathematical definition, $0! = 1$ (representing the single empty set arrangement).
3. **Legendre's Formula (Trailing Zeros):** Count factors of 5 in $n! \implies Z(n) = \sum_{k=1}^\infty \lfloor \n / 5^k \rfloor$.
4. **Stirling's Approximation:** For large $n$, $n! \approx \sqrt{2\pi n} \left(\n / e\right)^n$.
Formula Explanation
Your factorial calculations use product definitions and Stirling's equation:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for calculating 6!:
- Step 1 (Write Expression): $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$.
- Step 2 (Multiply First Pair): $6 \times 5 = \mathbf{30}$.
- Step 3 (Multiply Next Factors): $30 \times 4 = \mathbf{120}$.
- Step 4 (Continue Product): $120 \times 3 = \mathbf{360}$.
- Step 5 (Final Multiplication): $360 \times 2 \times 1 = \mathbf{720}$, delivering **6! = 720**!
Calculation Examples: Real-World Scenario Comparison
Compare factorial values, digit lengths, trailing zero counts, and Stirling estimates:
| Integer Input (n) | Calculated Factorial n! | Digit Count | Trailing Zeros | Stirling's Estimate |
|---|---|---|---|---|
| 0! | 1 | 1 digit | 0 zeros | 1.00 |
| 6! | 720 | 3 digits | 1 zero | 710.08 |
| 10! | 3,628,800 | 7 digits | 2 zeros | 3,598,695.62 |
| 25! | 15,511,210,043,330,985,984,000,000 | 26 digits | 6 zeros | 1.5459e+25 |
Benefits of Using the Factorial Calculator
Utilizing this calculator provides essential combinatorics, Taylor series, and probability advantages:
- BigInt Arbitrary Precision: Calculates exact factorials up to 1000! without truncation or scientific notation loss.
- Trailing Zeros Solver: Computes the exact count of trailing zeros using Legendre's formula.
- Combinatorics & Probability Core: Essential building block for permutations ($nPr$), combinations ($nCr$), and binomial expansions.
- Taylor Series & Calculus: Computes denominators for sine, cosine, and exponential $e^x$ series expansions.
Frequently Asked Questions (FAQ)
What is a factorial in math?
A factorial (denoted by n!) is the product of all positive integers less than or equal to n.
Why is 0! equal to 1?
0! = 1 by definition because there is exactly 1 way to arrange 0 items (an empty set). It also maintains algebraic consistency for n! = n × (n - 1)!.
Can negative numbers have factorials?
Standard factorials are defined only for non-negative integers. Negative integers do not have valid factorials (Gamma function has poles at negative integers).
What is Stirling's approximation?
Stirling's approximation (n! ≈ √(2πn) × (n/e)^n) estimates large factorials without performing all individual multiplications.
How do you find the number of trailing zeros in n!?
Count factors of 5 in n! using Legendre's formula: ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + ...
What is 5!?
5! = 5 × 4 × 3 × 2 × 1 = 120.
What is 10!?
10! = 3,628,800.
How fast does n! grow?
Factorial growth (O(n!)) is faster than exponential growth (O(2^n) or O(e^n)), making factorials grow extremely huge even for small n (e.g. 52! has 68 digits).
What is the Gamma function?
The Gamma function Γ(z) extends factorials to real and complex numbers, where Γ(n) = (n - 1)!.
What are subfactorials (!n)?
Subfactorials (!n) count derangements—permutations of n elements where no element appears in its original position.