Factorial Calculator
Calculate factorial instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods. Free to use with no signup required.
Calculator
Adjust values & calculateFactorial Growth Table
Formula
Where n! (n factorial) is the product of all positive integers from 1 to n. By convention, 0! = 1. Factorials count the number of ways to arrange n distinct objects in order (permutations).
Last reviewed: December 2025
Worked Examples
Example 1: Computing 8! with Trailing Zeros
Example 2: Factorials in Combinations
Background & Theory
The Factorial Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Factorial Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
n! = n * (n-1) * (n-2) * ... * 2 * 1
Where n! (n factorial) is the product of all positive integers from 1 to n. By convention, 0! = 1. Factorials count the number of ways to arrange n distinct objects in order (permutations).
Worked Examples
Example 1: Computing 8! with Trailing Zeros
Problem: Calculate 8! and determine how many trailing zeros it has.
Solution: 8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1\n= 56 * 6 * 5 * 4 * 3 * 2 * 1\n= 336 * 5 * 4 * 3 * 2 * 1\n= 1,680 * 4 * 3 * 2 * 1\n= 6,720 * 3 * 2 * 1\n= 20,160 * 2 * 1\n= 40,320\n\nTrailing zeros: floor(8/5) = 1\n8! = 40,320 has 1 trailing zero
Result: 8! = 40,320 | 1 trailing zero | 5 digits
Example 2: Factorials in Combinations
Problem: How many ways can you choose 3 items from 10? Use the combination formula with factorials.
Solution: C(10, 3) = 10! / (3! * 7!)\n= 3,628,800 / (6 * 5,040)\n= 3,628,800 / 30,240\n= 120\n\nAlternatively: (10 * 9 * 8) / (3 * 2 * 1) = 720 / 6 = 120
Result: C(10,3) = 10! / (3! * 7!) = 120 ways
Frequently Asked Questions
What is a factorial and how is it calculated?
A factorial of a non-negative integer n, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120. By mathematical convention, 0! is defined as 1, which is called the empty product. This definition is necessary to make many formulas in combinatorics and calculus work correctly. Factorials grow extremely rapidly: 10! = 3,628,800, 15! = 1,307,674,368,000, and 20! = 2,432,902,008,176,640,000. This explosive growth rate makes factorials one of the fastest-growing functions commonly encountered in mathematics. The factorial function is central to permutations, combinations, and probability theory.
Why is 0 factorial equal to 1?
The definition 0! = 1 follows logically from the recursive definition of factorials and from combinatorial reasoning. The recursive formula states n! = n * (n-1)!. If we apply this with n = 1: 1! = 1 * 0!, which gives 1 = 1 * 0!, so 0! must equal 1. From a combinatorics perspective, 0! counts the number of ways to arrange zero objects, and there is exactly one way to arrange nothing: do nothing. This is the empty permutation. Additionally, the binomial coefficient C(n, 0) = n! / (0! * n!) must equal 1 (there is one way to choose nothing), which requires 0! = 1. The gamma function, which extends factorials to non-integers, also gives the value 1 at the point corresponding to 0!.
How do you count trailing zeros in a factorial?
Trailing zeros in n! are produced by factors of 10, and each factor of 10 comes from pairing a factor of 2 with a factor of 5. Since factors of 2 are always more abundant than factors of 5 in n!, the number of trailing zeros equals the number of times 5 appears in the prime factorization of n!. This is calculated using Legendre formula: floor(n/5) + floor(n/25) + floor(n/125) + ... For example, 100! has floor(100/5) + floor(100/25) + floor(100/125) = 20 + 4 + 0 = 24 trailing zeros. The first term counts multiples of 5, the second counts multiples of 25 (which contribute an extra factor of 5), and so on for higher powers of 5.
What inputs do I need to use Factorial Calculator accurately?
Each field is labelled with the required unit (metric or imperial). Gather your source values before starting — for example, a weight measurement in kilograms, a distance in metres, or a dollar amount — and enter them exactly as measured. The formula section on this page lists every variable and explains what each represents.
How accurate are the results from Factorial Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
Does Factorial Calculator work offline?
Once the page is loaded, the calculation logic runs entirely in your browser. If you have already opened the page, most calculators will continue to work even if your internet connection is lost, since no server requests are needed for computation.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy