Factorial Calculator

A factorial calculator multiplies a whole number by every positive integer below it: n! = n × (n − 1) × … × 2 × 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Enter any non-negative integer to get the exact value of n!, its digit count, its trailing zeros and the full expansion. Exact big-integer math keeps even enormous inputs like 100! (a 158-digit number) precise to the last digit.

Factorials are the building block of counting. To arrange or choose only some of the items, the Permutation & Combination Calculator divides the factorials for you; to turn those counts into odds, use the Probability Calculator; for the coefficients of (a + b)ⁿ, the Binomial Expansion Calculator; and for other numbers too long for an ordinary calculator, the Big Number Calculator.

What a factorial actually means

Imagine you have three different books and want to line them up on a shelf. The first slot can hold any of the 3 books, the second slot any of the remaining 2, and the last slot the final 1 book. Multiply those choices together (3 × 2 × 1) and you get 6 possible orderings. That product is exactly 3!. In general, n! counts the number of distinct ways to arrange n different items in a row. This is why factorials are the backbone of combinatorics: permutations and combinations, and the probability formulas built on them, all rest on top of the humble factorial.

The factorial formula and definition

The factorial of a non-negative integer n is the product of every whole number from 1 to n:

There is also a neat recursive definition that says each factorial is built from the one before it:

Both definitions produce identical results. The recursive form is handy in programming and proofs, while the spelled-out product is easier to picture. The crucial base case is 0! = 1: there is exactly one way to arrange nothing (do nothing), and that value keeps every formula built on factorials consistent.

Worked example: 5!

Let's compute 5! by hand, multiplying one step at a time:

• Start at 1.
• 1 × 2 = 2
• 2 × 3 = 6
• 6 × 4 = 24
• 24 × 5 = 120

So 5! = 120. That means five different items can be ordered in 120 distinct ways: the same number of ways five guests can be seated in a row, or five runners can finish a race.

Factorial step by step: computing 8! as a running product

This is exactly how the calculator works internally and how you would do it on paper: start at 1 and multiply by each integer in turn, carrying the running total forward.

After the final step the running product is 8! = 40,320. The same loop scales to any n: each row simply multiplies the previous total by the next integer. (For large n the calculator groups the same multiplications into a tree of partial products, which gives the identical result much faster.)

How to use this factorial calculator

You only need one number to get a complete answer. Here is the workflow:

1. Enter n: type a non-negative whole number (for example 0, 7, 20, or 100) into the input box. A thousands separator such as 10,000 is fine.
2. Read the result: the large blue number at the top is n!. Up to 30 digits it is shown in full with thousands separators; longer results show a scientific-notation preview rounded to 6 significant digits, next to the digit count and the number of trailing zeros.
3. See the expansion: the "Expanded multiplication" card spells out n × (n − 1) × … × 1 so you can follow the logic.
4. Follow the steps: for n up to 12, the step-by-step table shows the running product after each multiplication, mirroring how you would do it on paper.
5. Expand huge results: when a factorial has hundreds of digits, open the "Show all digits" panel to reveal the exact integer.

The result updates instantly as you type; there is no "calculate" button to press. Use the quick-pick chips (0!, 5!, 10!, 13!, 20!, 52!, 100! and 1000!) to jump to common values. If you type a decimal such as 2.5, the calculator explains that factorials need whole numbers and shows the gamma function value Γ(3.5) instead.

Who this calculator is for

Factorials sit at the crossroads of school math and real-world counting problems, so this tool serves a wide audience:

• Students learning combinatorics, probability, or the binomial theorem who need to check homework.
• Teachers who want a clean expansion to project or share when introducing permutations.
• Programmers verifying a factorial function or testing against an exact reference value.
• Statisticians and data scientists computing permutations, combinations, or Poisson-style terms by hand.
• Anyone curious about just how fast factorials explode: try 70! and watch the digit count climb past 100.

Worked example: 10! and a permutation

Computing 10! gives 3,628,800, the number of ways to order ten distinct items. Now suppose you only care about arranging 3 of those 10 items in order. That is a permutation, written P(10, 3), and it equals 10! / (10 − 3)! = 10! / 7! = 3,628,800 / 5,040 = 720. Notice how the factorial does the heavy lifting: you compute the full count and then divide away the orderings you do not care about. The same idea drives combinations, where you divide by an extra factorial to ignore order entirely. If you need those counts directly, the Permutation & Combination Calculator does the division for you.

Factorials behind common permutations and combinations

Most real counting problems are permutations P(n, r) = n! / (n − r)! (order matters) or combinations C(n, r) = n! / (r! × (n − r)!) (order does not). Both are just factorials divided by factorials. This table shows the exact counts for cases people look up most.

Every value here comes straight from factorials: for the 5-card poker hand, C(52, 5) = 52! / (5! × 47!) = 2,598,960 possible hands.

Worked example: a huge factorial

Enter 52 and you get the number of ways to shuffle a standard deck of cards: 52! ≈ 8.07 × 10^67 (8.06582 × 10^67 to six digits), a 68-digit number. That figure is so vast that a thoroughly shuffled deck is almost certainly in an order no deck has ever been in before. Push to 100! and the result has 158 digits, the last 24 of them zeros. Ordinary calculators give up or round here, but this tool keeps every digit because it uses arbitrary-precision integers; expand the digits panel to see all 158 of them.

How many digits does a large factorial have?

For big inputs the exact integer is impractical to read, so the useful facts are its digit count and its scientific-notation form. This table lists both for the values people search for most, from 20! to 1000!, rounded to three significant digits.

Around n = 170 a standard double-precision float overflows to infinity, since 171! (about 1.24 × 10^309) exceeds the largest representable double, about 1.8 × 10^308. That is the point where you must switch to big-integer or log-based math. This calculator does both: BigInt for exact values up to 20,000!, and the log-gamma function for the estimates beyond.

Factorial reference table

Here are the factorials of 0 through 20, a handy reference for spot-checking and seeing how quickly they grow:

Notice that 13! already exceeds 6 billion, and by 20! the value passes 2.4 quintillion. This explosive growth is why factorials outrun exponential functions and why even modest inputs need big-integer math.

Key terms explained

• Factorial (n!): the product of all positive integers up to n; counts orderings of n items.
• Permutation P(n, r): the number of ordered arrangements of r items from n, equal to n! / (n − r)!.
• Combination C(n, r): the number of unordered selections of r from n, equal to n! / (r! × (n − r)!).
• Gamma function Γ: a continuous extension of the factorial where Γ(n + 1) = n!, defined for non-integers too.
• Double factorial (n!!): the product of integers with the same parity as n (for example 7!! = 7 × 5 × 3 × 1); a different operation from the ordinary factorial.
• Stirling's approximation: a formula, n! ≈ √(2πn) (n/e)^n, that estimates large factorials without multiplying every term.

Why factorials grow so fast

Each step in a factorial multiplies by a larger number than the last, so the value compounds at an ever-increasing rate. An exponential like 2^n multiplies by the same factor (2) each step; a factorial multiplies by n, n − 1, n − 2 and so on. That is why n! eventually overtakes any fixed-base exponential and why the number of digits climbs steeply. Stirling's approximation captures this: it shows the digit count grows roughly in proportion to n log n, which is why 100! lands at 158 digits and 1000! at over 2,500.

Where factorials show up

Beyond textbook exercises, factorials appear constantly:

• Card games and shuffling: 52! is the number of possible deck orders.
• Probability and odds: permutations and combinations, both built on factorials, count favorable and total outcomes.
• The binomial theorem: binomial coefficients "n choose r" use factorials to expand (a + b)^n.
• Calculus and series: Taylor series for sine, cosine and e^x divide each term by a factorial.
• Algorithms: the brute-force cost of checking every ordering (such as every route in the traveling-salesman problem) grows like n!.

Tips for working with factorials

• Cancel before you multiply: in expressions like 10! / 7!, cancel the shared terms (10! / 7! = 10 × 9 × 8 = 720) instead of computing both factorials.
• Remember 0! = 1 and 1! = 1: these base cases trip up many beginners.
• Use exact arithmetic: ordinary double-precision numbers hold n! exactly only through 22! and round the trailing digits from 23! on; this calculator's BigInt math does not.
• Estimate with Stirling: when you only need the rough size or digit count, Stirling's approximation is faster than the full product.

How it compares to related calculators

This page answers "what is n!?" If your question is a bit different, a sister tool fits better:

• To count ordered or unordered selections directly (nPr and nCr), use the Permutation & Combination Calculator.
• To turn those counts into odds and chances, use the Probability Calculator.
• To raise a number to a power, use the Exponent Calculator.
• To expand (a + b)ⁿ with its binomial coefficients, use the Binomial Expansion Calculator.
• For exact arithmetic on other very long numbers, use the Big Number Calculator.
• For another fast-growing integer sequence, see the Fibonacci Calculator.
• To write a giant factorial like 100! in the form a × 10^b, use the Scientific Notation Calculator.
• For trig, logs and general expressions, use the Scientific Calculator.

Factorials in computing and notation

Factorials are a favorite teaching example in programming because they show off two ideas at once: iteration (a loop that multiplies a running total by 1, 2, 3, …, n) and recursion (a function that calls itself with n − 1 until it hits the base case 0! = 1). The catch is size. A standard 64-bit integer overflows at 21!, and a double-precision floating-point number stops being exact at 23! (it holds 22! exactly only because factorials end in many binary zeros), so any serious factorial routine needs arbitrary-precision (big-integer) arithmetic, which is exactly what this calculator uses under the hood. When a result has too many digits to read comfortably, it is conventional to write it in scientific notation, for example 100! ≈ 9.33 × 10¹⁵⁷, while keeping the full exact integer available on demand.

A quick note on notation you may meet alongside the plain factorial. The double factorial n!! multiplies only every second term down from n (so 8!! = 8 × 6 × 4 × 2 = 384), which is a different operation, not "the factorial of a factorial." The subfactorial !n counts derangements (permutations where nothing stays in its original place), and the falling factorial appears in the permutation formula. All of these descend from the same counting idea, but only the ordinary single factorial is what this page computes.

Frequently asked quick facts

• What is 5! ? 5! = 120.
• What is 6! ? 6! = 720.
• What is 10! ? 10! = 3,628,800.
• What is 12! ? 12! = 479,001,600, the largest factorial that still fits in a 32-bit signed integer.
• What is 13! ? 13! = 6,227,020,800, the first factorial that overflows a 32-bit integer.
• What is 0! ? 0! = 1, by definition.
• What is 1! ? 1! = 1.
• How many digits does 100! have? 158 digits, ending in 24 zeros.
• What is 0.5! ? Not an ordinary factorial; the gamma function gives Γ(1.5) = √π ÷ 2 ≈ 0.886227.

These are the values people most often look up, and they double as a fast sanity check: a tool that returns 0 for 0! has the definition wrong, and one that rounds the trailing digits of 25! is using floating-point math rather than the exact big-integer arithmetic this calculator relies on.

Sources & further reading

• NIST Digital Library of Mathematical Functions, §5.4 Special Values and Extrema: n! = Γ(n + 1) (5.4.1) and Γ(1/2) = √π (5.4.6).
• NIST Digital Library of Mathematical Functions, §5.2 Definitions: the gamma function has simple poles at z = 0, −1, −2, …, which is why negative whole numbers have no factorial.
• NIST Digital Library of Mathematical Functions, §5.11 Asymptotic Expansions: Stirling's series for ln Γ(z), used for the estimates above 20,000!, and Stirling's formula n! ≈ √(2πn) (n/e)ⁿ.
• ECMAScript Language Specification, §6.1.6.2 The BigInt Type (integers of any size) and §21.1.2.7 Number.MAX_VALUE (about 1.8 × 10³⁰⁸, the reason ordinary numbers overflow after 170!).