Factorial Calculator

Factorials hide inside an surprising amount of everyday mathematics — counting arrangements, computing probabilities, and expanding functions into series all depend on them. The factorial calculator takes any non-negative integer and returns the product of every whole number from that integer down to 1, instantly and exactly, even when the answer grows to dozens of digits. It is the quickest way to check hand work on counting problems or to grab a value like 15! without wearing out a pencil.

Quick Answer

n! = n × (n − 1) × (n − 2) × … × 2 × 1, with the special case 0! = 1. So 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. The exclamation mark is the factorial symbol, read aloud as "n factorial."

Why Factorials Show Up Everywhere

The factorial counts arrangements. If six runners finish a race with no ties, there are 6! = 720 possible finishing orders, because any of the six can win, leaving five choices for second, four for third, and so on. That counting logic is the backbone of permutations and combinations, which is why factorials dominate probability chapters: lottery odds, card hands, seating plans and password spaces are all built from them.

Factorials also grow faster than almost anything else in school mathematics. Powers like 2ⁿ feel steep until you compare: 10! is already 3,628,800, and 20! passes 2.43 × 10¹⁸. This explosive growth is exactly why brute-forcing a shuffled deck is hopeless — 52! is about 8 × 10⁶⁷, so a random shuffle has almost certainly never been repeated in history. In higher mathematics the same products appear in the series for eˣ, sine and cosine, where each term divides by a factorial, taming the growth and letting the series converge. The definition extends to non-integers through the gamma function, but the calculator works with the standard whole-number definition used in counting and probability.

The odd-looking rule 0! = 1 is not arbitrary. It makes the counting formulas consistent — there is exactly one way to arrange zero objects — and it keeps the pattern n! = n × (n − 1)! working all the way down: 1! = 1 × 0! forces 0! to equal 1.

How to Use the Factorial Calculator

1. Choose a non-negative whole number — factorials in this standard form are not defined for negatives or fractions.
2. Enter the number into the input field.
3. Read the result; for large inputs expect a very long number or scientific notation.
4. If you are working a permutations problem, note the factorial values you will need before combining them.
5. Sanity-check small results by hand: anything up to 5! (120) is quick to verify mentally.

Building 6! by Hand — and Watching It Take Off

Start from the definition and multiply the chain out: 6! = 6 × 5 × 4 × 3 × 2 × 1. Step by step that is 6 × 5 = 30, 30 × 4 = 120, 120 × 3 = 360, 360 × 2 = 720, and multiplying by 1 leaves 720. Each step reuses the previous factorial — 4! = 24 sits inside 5! = 120, which sits inside 6! — which is why the values escalate so violently.

The first stretch of the sequence is worth having at your fingertips:

• 0! = 1 and 1! = 1
• 2! = 2, 3! = 6, 4! = 24, 5! = 120
• 7! = 5,040 and 8! = 40,320
• 10! = 3,628,800

Limits and Slip-Ups With Factorials

The classic slip is arithmetic drift in the middle of the chain: one wrong partial product poisons everything after it, which is the practical case for using a calculator on anything above 7!. A conceptual trap is the double factorial, written n!! — it multiplies only every second number (7!! = 7 × 5 × 3 × 1 = 105) and is not the factorial of a factorial. Another is assuming factorials behave politely with addition: (3 + 2)! is 120, while 3! + 2! is just 8, so brackets matter enormously.

Overflow is the honest limitation. Factorials outgrow ordinary calculator displays quickly — 25! is roughly 1.55 × 10²⁵ — and outgrow exact integer arithmetic in many programs soon after. When a problem only needs a ratio, such as a probability, the factorials often cancel; forming the fraction before computing can keep numbers manageable. And remember the domain: negative integers have no factorial in this system, so an input of −3 is an error, not a puzzle.

Factorials Inside a Real Count: Poker Hands

Watch factorials do a full job. How many different five-card hands can be dealt from a standard 52-card deck? Choosing 5 cards from 52, order irrelevant, gives 52! ÷ (5! × 47!). The factorials look monstrous, but almost everything cancels: after cancelling 47!, the numerator reduces to 52 × 51 × 50 × 49 × 48 = 311,875,200, and the denominator is just 5! = 120. Divide, and the answer is 311,875,200 ÷ 120 = 2,598,960 possible hands. This cancellation is the standard technique — never compute the giant factorials separately when a ratio is involved, because the shared tail divides away. It also explains the feel of card games: nearly 2.6 million possible hands is why two players are essentially never dealt identical combinations, and why probabilities for specific hands, like a royal flush at 4 in 2,598,960, come out so small.

When a factorial is too large to compute exactly but an estimate will do, mathematicians reach for Stirling's approximation, which says n! behaves like √(2πn) × (n/e)ⁿ for large n. It is startlingly good even early on: applied to 10! it gives roughly 3.6 million against the true 3,628,800 — within about one percent — and the relative error only shrinks as n grows. It is the standard way to estimate enormous factorials in probability and statistical mechanics without ever writing them out.

Putting Factorials to Work

Once factorials are effortless, the counting tools built on them open up. The permutation and combination calculator applies factorials to ordered and unordered selections, the combination calculator focuses on choosing groups, and the probability calculator turns those counts into odds. You will find all of them alongside this one in the math calculators collection.