AkCalculators

Factorials explained — What is n!, how to compute it, examples and practical uses

Factorial is one of those simple mathematical operations that appears everywhere: in counting, probability, combinatorics, calculus (series expansions), and algorithms. The factorial of a non-negative integer n, written n!, is the product of all positive integers up to n: n! = 1 × 2 × 3 × ... × n. By convention 0! = 1. Although the definition is straightforward, factorials grow very rapidly and invite interesting computational and theoretical questions. This article blends practical tutorial guidance (how to use this calculator) with educational background about factorials, their properties, algorithms for computation, and real-world applications.

Definition and quick examples

Definition: For integer n ≥ 1, n! = n × (n−1) × (n−2) × ... × 1. Special case: 0! = 1.

Examples:

• 1! = 1
• 2! = 2
• 3! = 6
• 5! = 120
• 10! = 3,628,800

Why 0! = 1?

Defining 0! as 1 keeps many combinatorial formulas consistent. For instance, the number of ways to choose 0 objects from n is C(n,0) = n!/(0!(n−0)!) = 1, so 0! must equal 1. Similarly, empty products in algebra are usually taken to be 1, which aligns with factorial conventions.

Combinatorics: permutations and combinations

Factorials are the backbone of counting formulas. The number of permutations of n distinct items is n!. If you want permutations of k items from n (order matters), the count is P(n,k) = n!/(n−k)!. For combinations (order does not matter), C(n,k) = n! / (k!(n−k)!). These formulas make factorial computations essential in probability, statistics, and algorithm analysis.

Computing factorials in practice

The simplest and most common algorithm to compute n! is iterative multiplication using arbitrary-precision integers (BigInt in JavaScript, GMP in C/C++). For typical inputs (n up to a few thousands), straightforward iteration is fine. For extremely large n (tens or hundreds of thousands), specialized algorithms accelerate multiplication using divide-and-conquer, Fast Fourier Transform (FFT)-based multiplication and prime-swing or binary-splitting factorial algorithms implemented in high-performance libraries.

Using this calculator — step-by-step

1. Enter a non-negative integer in the input field. Example: enter 10 to compute 10!.
2. Optionally tick Show step-by-step to view the multiplication expansion (for large n this will be truncated to limit page load).
3. Set the Max terms to display if you want a shorter or longer step preview.
4. Click Calculate. The tool uses BigInt to compute n! exactly and displays the result, the number of digits, and timing metadata.
5. Use Download CSV to save the numeric result and metadata, or Copy Result to copy a summary to clipboard.

Interpreting outputs and metadata

After computation the page shows:

• Value: exact n! as an integer (may be very long).
• Digits: number of decimal digits in n! (useful to estimate size).
• Computation time: approximate elapsed time in milliseconds.
• Step preview: multiplication string or truncated snippet for large n.

Approximations and Stirling’s formula

Because factorials grow so fast, approximations are useful. Stirling’s approximation is a widely used formula:

n! ≈ √(2πn) (n/e)^n

This gives a good estimate of the magnitude and number of digits of n!. A refined expansion gives even more accuracy, and many algorithms use logarithms of factorials (log n!) to avoid computing the full number when only magnitude matters.

Performance and browser limits

Computing n! for n up to a few thousand is usually fine in modern browsers using BigInt. For n in the tens of thousands, the resulting integer has hundreds of thousands of digits and can be slow or exceed memory limits. This calculator truncates step-by-step output if n exceeds your chosen display limit and warns if n is extremely large. For production or research needs, use server-side big-integer libraries (GMP, Python's gmpy2) or specialized factorial algorithms.

Applications beyond counting

Factorials appear in series expansions (Taylor series for e^x, sin, cos), probability distributions (Poisson, permutations), combinatorics, and algorithm complexity analysis. In calculus and complex analysis the Gamma function extends factorials to non-integer values via Γ(n+1)=n! and is central to advanced mathematical modeling.

Example use cases

• Compute permutations: number of ways to arrange 8 distinct objects = 8! = 40,320.
• Compute combinations: choose 5 from 20 = 20!/(5!15!).
• Estimate sizes: number of digits in 100! is floor(log10(100!)) + 1 ≈ 158.

Summary

Factorials are deceptively simple to define but can produce enormous numbers quickly. This Factorial Calculator provides exact results using BigInt, step previews for learning, and export options for documentation. Use it for homework, quick checks, and moderate-size computations, and switch to high-performance libraries for extremely large tasks.