What is a Factorial (n!)?

The factorial of a non-negative integer n is the product of all positive integers less than or equal to n: n! = n × (n - 1) × (n - 2) × ... × 2 × 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Why is 0! equal to 1?

0! = 1 by mathematical definition to preserve recursive identities like n! = n × (n - 1)!, which for n = 1 gives 1! = 1 × 0!, requiring 0! = 1. Combinatorially, there is exactly 1 way to arrange an empty set of zero items.

How do you count trailing zeros in a factorial using Legendre's Formula?

Trailing zeros are formed by pairs of 2s and 5s (since 2 × 5 = 10). Because factors of 2 are abundant, count factors of 5: Zeros = floor(n/5) + floor(n/25) + floor(n/125) + ... For 100!: floor(100/5) + floor(100/25) = 20 + 4 = 24 trailing zeros.

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Factorial Calculator (n!) Studio

Step-by-Step Breakdown • Expressions Solver • Permutations & Combinations

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Factorial & Expression Solver

Calculate exact factorials, solve equations (e.g. 5! + 3! or 10!/5!), or use the slider

Enter Integer or Expression:5!

Type an integer (e.g. 10), a factorial notation (10!), or complex formulas.

Quick Explore Slider (0 - 170):5

Drag to instantly compute exact factorials from 0! to 170!.

Result (5!):

120

Step-by-Step Multiplication

5! = 5 × 4 × 3 × 2 × 1

= 5 × 4!

= 20 × 3!

= 60 × 2!

= 120 × 1

= 120

Mathematical Presets & Benchmarks

Quickly load specific mathematical constants, arrangements, and combinatorics limits

Arrangements (nPr) & Selections (nCr)

Calculate probabilities, permutations, and choices using optimized overflow-proof algorithms

Permutations (nPr)

n! / (n - r)!

Total Items (n):

Arranged (r):

10P3 =720

Order matters (e.g. race finishes, passcode digits).

Combinations (nCr)

n! / [r! × (n - r)!]

Total Items (n):

Chosen (r):

10C3 =120

Order does not matter (e.g. lottery picks, team formations).

Stirling's Approximation Studio

Estimate giant factorials up to 10,000! and track mathematical error tolerances

Enter n for Estimation (0 - 10,000):

Stirling's theorem calculates $\ln(n!) \approx n\ln(n) - n + 0.5\ln(2\pi n)$ to bypass double-precision overflow limits.

Error Matrix (Exact vs Stirling)

Target Factorial:	5!
Exact Value:	120
Stirling's Estimate:	1.180 × 10^2
Absolute Error:	1.98
Relative Margin:	1.6500%

💡 Margin drops below **1%** at $n \ge 10$. As $n \to \infty$, relative error approaches zero.

Recent Tape Calculations

combination 10C3→120

permutation 10P3→720

factorial 5!→120

Factorial Reference Chart (0! to 20! Quick Guide)

Instant numerical guides, digit counts, and landmark associations from 0! to 20!

A factorial, written as n!, is the product of all positive integers from 1 up to and including n. By mathematical convention, 0! = 1 because there is exactly one way to arrange zero items — the empty arrangement. Factorials grow extremely rapidly: 10! is already 3,628,800, while 20! exceeds 2.4 quintillion (2,432,902,008,176,640,000). This explosive growth makes factorials central to probability theory, combinatorics, and algorithm complexity.

n	n! (Factorial Value)	Digit Count	Common Association
0	`1`	1	Empty set permutation (by definition)
1	`1`	1	Single element — only one arrangement
2	`2`	1	Two items: AB or BA
3	`6`	1	RGB colour orderings
4	`24`	2	Card suit orderings
5	`120`	3	5-character PIN permutations
6	`720`	3	Chess piece movement orderings
7	`5,040`	4	Days-of-week arrangement count
8	`40,320`	5	Rubik's Cube corner arrangements
9	`362,880`	6	Sudoku digit arrangements
10	`3,628,800`	7	10-player team lineup permutations
11	`39,916,800`	8	11-player football formation orders
12	`479,001,600`	9	12-month calendar arrangement count
13	`6,227,020,800`	10	Standard deck suit orderings (partial)
14	`87,178,291,200`	11	Cryptographic key space baseline
15	`1,307,674,368,000`	13	Tournament bracket permutations
16	`20,922,789,888,000`	14	16-bit register state orderings
17	`355,687,428,096,000`	15	Genome sequencing combinatorics
18	`6,402,373,705,728,000`	16	Chessboard piece placements
19	`121,645,100,408,832,000`	18	Protein folding sequence counts
20	`2,432,902,008,176,640,000`	19	Upper limit for 64-bit integer storage

💡 Practical Mathematics

Real-World Applications of Factorials

How factorial mathematics powers cryptography, algorithm design, probability modeling, and modern scientific computing.

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P(n, k) = n! ÷ (n − k)!

Permutations — Ordered Arrangements

A permutation counts the number of ways to arrange k items selected from n distinct objects where order matters. The formula is P(n, k) = n! ÷ (n − k)!. If a race has 10 competitors and you want to know how many possible 1st-2nd-3rd podium combinations exist, that is P(10, 3) = 10! ÷ 7! = 720. Permutations appear in cryptography (counting possible password sequences), scheduling (task ordering in operating systems), and genetics (counting codon sequence arrangements in RNA).

🔒 Cryptography⏱️ OS Scheduling🧬 RNA Genetics

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C(n, k) = n! ÷ (k! × (n − k)!)

Combinations — Unordered Selections

A combination counts the number of ways to choose k items from n objects where order does not matter. The formula is C(n, k) = n! ÷ (k! × (n − k)!), also written as "n choose k". Selecting 5 lottery numbers from 49 gives C(49, 5) = 1,906,884 possible tickets. Combinations appear in probability distributions (binomial theorem), card game hand counting, and statistical sampling design where only which items are selected matters, not their sequence.

🎟️ Lottery Odds🃏 Card Games📊 Statistical Sampling

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eˣ = Σ (xⁿ ÷ n!)

Taylor Series & Calculus

Factorials appear as denominators in Taylor series expansions, which approximate mathematical functions as infinite polynomials. The exponential function e^x equals the sum of x^n ÷ n! for n from 0 to infinity. The sine and cosine functions, natural logarithm, and virtually every transcendental function used in physics and engineering is computed via factorial-denominator Taylor series internally by calculators and programming language math libraries.

📈 Taylor Series〰️ Trigonometry⚙️ Calculator Engines

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O(n!) Complexity

Computer Science — Algorithm Complexity

Factorial time complexity, written O(n!), is the worst-case growth rate of brute-force algorithms that enumerate every possible permutation of the input. The Travelling Salesman Problem (finding the shortest route visiting n cities) has O(n!) naive complexity because checking every possible route ordering requires n! iterations. For n = 15 cities, that is over 1.3 trillion route checks — illustrating why heuristic algorithms are necessary for even modest problem sizes. Understanding factorial growth is essential for algorithm design and NP-hard problem analysis.

🗺️ Travelling Salesman🧩 NP-Hard Problems💻 Algorithm Design

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P(X=k) = C(n,k)·pᵏ(1−p)ⁿ⁻ᵏ

Probability — Binomial Distribution

The binomial distribution, which models the probability of getting exactly k successes in n independent trials (such as coin flips or quality control pass/fail tests), uses the combination formula C(n, k) directly. Calculating the probability that exactly 3 out of 10 components fail a stress test uses 10! in its numerator. Epidemiologists use factorial-based formulas to model disease spread probabilities, clinical trial statisticians use them for significance testing, and insurance actuaries use them for mortality rate modelling.

🦠 Epidemiology🧪 Clinical Trials📋 Actuarial Science