Arithmetic
Factorial Calculator
Compute n! exactly — no 170! ceiling — with the full expansion, digit count and trailing zeros. Switch tabs for permutations, combinations, double factorials and derangements.
Factorial
n!5! =
120
Breakdown
5! = 5 × 4 × 3 × 2 × 1
digits
3
trailing 0s
1
input
n=5
Exact to the last digit — no 170! ceiling. Try n = 100 or more.
Try an example
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The basics
What is a factorial?
The factorial of a whole number multiplies that number by every whole number beneath it, down to 1. It is the answer to a single, surprisingly common question: in how many different orders can I arrange these items? That makes it the foundation of counting, probability, and the permutation and combination formulas.
n! = n × (n − 1) × … × 2 × 1
The definition
Multiply every whole number from n down to 1. The product is the factorial — the count of ways to line n distinct items up in a row.
5! = 5 × 4 × 3 × 2 × 1 = 120
0! = 1
The empty-set convention
There is exactly one way to arrange nothing: do nothing. Defining 0! this way keeps every downstream formula, like permutations and combinations, working without special cases.
1! = 1 and 0! = 1
Method
How to calculate a factorial by hand
Three steps turn any small factorial into a number you can check.
- 1
Write the descending product
Start at n and write every whole number beneath it, joined by multiplication, ending at 1.
6! = 6 × 5 × 4 × 3 × 2 × 1 - 2
Multiply left to right
Keep a running product. Multiply the first two numbers, then carry that total into the next, and so on.
6×5 = 30 → ×4 = 120 → ×3 = 360 - 3
Read off the result
Finish the chain down to 1. The final running total is n!.
… × 2 × 1 = 720, so 6! = 720
Worked examples
Factorials, permutations, combinations
The same factorial machinery answers three different counting questions. Each tab works one through, computed live from the formula.
Given Find 6!
Write the product: 6! = 6 × 5 × 4 × 3 × 2 × 1
Multiply left to right: 6 × 5 = 30, × 4 = 120, × 3 = 360, × 2 = 720
6! = 720
Quick chart
Factorial table, 0! to 25!
A ready reference for the small factorials people look up most — with each exact value, its scientific form, how many digits it has, and how many zeros it ends in.
| n | n! (exact) | Scientific | Digits | Trailing 0s |
|---|---|---|---|---|
| 0! | 1 | — | 1 | 0 |
| 1! | 1 | — | 1 | 0 |
| 2! | 2 | — | 1 | 0 |
| 3! | 6 | — | 1 | 0 |
| 4! | 24 | — | 2 | 0 |
| 5! | 120 | — | 3 | 1 |
| 6! | 720 | — | 3 | 1 |
| 7! | 5,040 | — | 4 | 1 |
| 8! | 40,320 | — | 5 | 1 |
| 9! | 362,880 | — | 6 | 1 |
| 10! | 3,628,800 | 3.62880e6 | 7 | 2 |
| 11! | 39,916,800 | 3.99168e7 | 8 | 2 |
| 12! | 479,001,600 | 4.79001e8 | 9 | 2 |
| 13! | 6,227,020,800 | 6.22702e9 | 10 | 2 |
| 14! | 87,178,291,200 | 8.71782e10 | 11 | 2 |
| 15! | 1,307,674,368,000 | 1.30767e12 | 13 | 3 |
| 16! | 20,922,789,888,000 | 2.09227e13 | 14 | 3 |
| 17! | 355,687,428,096,000 | 3.55687e14 | 15 | 3 |
| 18! | 6,402,373,705,728,000 | 6.40237e15 | 16 | 3 |
| 19! | 121,645,100,408,832,000 | 1.21645e17 | 18 | 3 |
| 20! | 2,432,902,008,176,640,000 | 2.43290e18 | 19 | 4 |
| 21! | 51,090,942,171,709,440,000 | 5.10909e19 | 20 | 4 |
| 22! | 1,124,000,727,777,607,680,000 | 1.12400e21 | 22 | 4 |
| 23! | 25,852,016,738,884,976,640,000 | 2.58520e22 | 23 | 4 |
| 24! | 620,448,401,733,239,439,360,000 | 6.20448e23 | 24 | 4 |
| 25! | 15,511,210,043,330,985,984,000,000 | 1.55112e25 | 26 | 6 |
Where it shows up
Why factorials matter
A simple descending product that quietly underpins probability, computer science, and calculus.
52! orderings
Probability & cards
The number of ways to shuffle a deck is 52! — a 68-digit figure so large that every shuffle you make has almost certainly never occurred before.
O(n!) time
Computer science
Brute-forcing every order of n items — the travelling-salesman tour, say — takes n! steps, which is why such problems get hard so fast.
eˣ = Σ xⁿ / n!
Calculus & series
Factorials sit in the denominators of Taylor series, letting polynomials approximate eˣ, sine, and cosine to any precision.
Worth knowing
Factorial rules worth remembering
A handful of identities and edge cases that make factorials easier to reason about.
n! = n × (n − 1)!
The recursive rule
Every factorial contains the one below it. This recurrence is how factorials are usually defined and computed in code.
0! = 1! = 1
The two base cases
Both the empty arrangement and the single-item arrangement have exactly one ordering, so both equal 1. These anchor the recursion.
negative n: undefined
No negatives or decimals
You cannot arrange a negative or fractional number of items, so the plain factorial stops at the non-negative whole numbers.
n!! ≠ (n!)!
Double factorial is different
A double factorial skips every other term going down; it is not the factorial applied twice. Mind the notation.
trailing 0s = Σ⌊n/5ᵏ⌋
Counting trailing zeros
Zeros come from pairing factors of 2 and 5; the 5s are scarcer, so they alone determine how many zeros n! ends in.
n! ≈ √(2πn)(n/e)ⁿ
Stirling's approximation
For large n, this formula estimates a factorial closely without multiplying everything out — handy in statistics.
Variants
Beyond the plain factorial
Several relatives extend the factorial idea — each available as a tab in the tool above.
nPr = n! ÷ (n − r)!
Permutations
Counts ordered arrangements of r items chosen from n. Order matters, so it is larger than the matching combination.
10P3 = 720
nCr = n! ÷ (r!(n − r)!)
Combinations
Counts unordered selections — the same items in a different order count once. These are the binomial coefficients.
52C5 = 2,598,960
n!! = n(n − 2)(n − 4)…
Double factorial
Multiplies every other number down from n. It appears in integration formulas and the volume of high-dimensional spheres.
7!! = 7 × 5 × 3 × 1 = 105
!n derangements
Subfactorial
Counts arrangements where no item lands in its own place — the “hat-check” problem. It hovers near n! ÷ e.
!4 = 9
Pitfalls
Common factorial mistakes
Three slips that trip people up most often.
0! = 1, not 0
Forgetting the convention
It is tempting to read 0! as zero. By definition it is 1 — the lone way to arrange nothing.
(2 + 3)! ≠ 2! + 3!
Factorials don't distribute
You cannot split a factorial across addition. Evaluate the inside first, then take the factorial of the result.
170! overflows doubles
Trusting a capped tool
Calculators that use ordinary floating point silently fail past 170!. For exact large values you need arbitrary precision — which this tool uses.
FAQ
Common questions
The factorial of a non-negative whole number n, written n!, is the product of every whole number from n down to 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials count the number of ways to arrange a set of items in order.
0! equals 1 by convention. A factorial counts the ways to arrange a set, and there is exactly one way to arrange an empty set — do nothing — so 0! = 1. This also keeps the formulas for permutations and combinations consistent (for example nP n = n! / 0! = n!).
Each step multiplies by a larger number, so the result compounds. 10! is already 3,628,800, and 70! exceeds the number of atoms in the observable universe. This explosive growth is why factorials dominate the running time of brute-force 'try every order' algorithms.
The basic factorial is defined only for non-negative whole numbers, because it counts whole arrangements of a set. The gamma function extends the idea to decimals and complex numbers (n! = Γ(n+1)), but it is undefined at zero and the negative integers.
Permutations (nPr = n! / (n − r)!) count ordered arrangements, where the sequence matters. Combinations (nCr = n! / (r!(n − r)!)) count unordered selections, where order is ignored. Both are built from factorials — switch the mode tabs above to compute either.
Count the factors of 5 in the numbers up to n: ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + … (Legendre's formula). 100! ends in 24 zeros, for instance, because zeros come from pairing 2s and 5s, and 5s are the scarcer factor.
There is no fixed ceiling. Most factorial calculators stop at 170! because that is the largest value a standard floating-point number can hold. This tool uses arbitrary-precision (BigInt) arithmetic, so it returns the exact, complete digits of values far beyond that — try 200! or 500!.
A double factorial multiplies every other number down from n: n!! = n · (n − 2) · (n − 4) · … For 7!! that is 7 × 5 × 3 × 1 = 105. It is not the factorial of a factorial — switch to the n!! tab to compute it.
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Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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