Factorial Calculator

A factorial, written as n!, is the product of every positive integer from 1 up to n. It is one of the fundamental building blocks of combinatorics and probability, because it counts the number of ways to arrange a set of distinct items. For example, five books can be arranged on a shelf in 5! = 120 different orders.

Factorials grow extremely quickly. 10! is already 3,628,800 and 20! passes two quintillion, which is why they are usually written in scientific notation or computed with arbitrary-precision arithmetic. A standard 64-bit integer overflows after 20!, so this calculator uses JavaScript's BigInt type to keep every digit exact.

Beyond plain factorials, this page also computes permutations (nPr) and combinations (nCr). Permutations count ordered arrangements, while combinations count unordered selections. The two appear constantly in statistics, lottery odds, card-game probabilities, and password-strength estimates.

Key Properties of Factorials

• Recurrence — every factorial is built from the one before it: n! = n × (n − 1)!. This is what makes iterative computation so cheap.
• Super-exponential growth — n! eventually outgrows any fixed exponential aⁿ, which is why factorials blow up faster than almost any function you will meet in everyday maths.
• Divisibility — n! is divisible by every integer from 1 to n by construction, a fact that underpins many number-theory proofs.
• Combinatorial meaning — n! counts the number of distinct orderings (permutations) of n distinct objects.
• Extension beyond integers — through the Gamma function, n! = Γ(n + 1), so factorials can be generalised to non-integer values.
• Exactness — because the values are whole numbers, they can be computed exactly with big-integer arithmetic rather than approximated in floating point.

1. Calculate factorial — Enter a non-negative integer (max 5000) and click Calculate.
2. View steps — See the step-by-step multiplication for smaller numbers.
3. Permutations & Combinations — Enter n and r values to calculate nPr and nCr instantly.

Factorials sit underneath a surprising number of everyday calculations. Whenever you need to count the ways something can be arranged or chosen, a factorial is usually not far away.

1. Counting arrangements — the number of ways to order n distinct items is exactly n!. Seating guests, ordering a playlist, or arranging books on a shelf are all factorial problems.
2. Selections and groups — choosing a committee or a poker hand uses combinations (nCr), which are built from factorials divided by other factorials.
3. Probability — event probabilities in games, lotteries, and the classic birthday problem are computed with factorial ratios.
4. Keyspace estimation — the number of orderings of a character set gives an upper bound on brute-force attempts, which is useful for assessing password and PIN strength.
5. Series and calculus — the Taylor series for e^x, sine, and cosine all use factorials in their denominators, so they appear throughout numerical analysis.
6. Algorithm analysis — brute-force algorithms over permutations, such as a naive travelling-salesman search, have a factorial running time that grows explosively with n.

The calculator multiplies 1 × 2 × 3 × … × n step by step, but instead of ordinary numbers it uses JavaScript's BigInt type. That means every digit of the result is stored exactly — including 1000!, which has 2,568 digits — with no rounding or scientific notation.

For permutations and combinations the tool reuses those exact big integers. Permutations use nPr = n! / (n − r)! and combinations use nCr = n! / (r! (n − r)!). Because both values are exact, the displayed results are correct to the last digit, even for inputs such as 52C5.

For a rough estimate of how large a factorial will be, you can use Stirling's approximation or the digit count formula. The number of digits in n! is approximately n × log10(n) − n × log10(e) + …, which is why the calculator reports the exact digit count alongside each result.