Factorial Calculator
Work out n! exactly, with every digit, the number of digits and the zeros at the end. Also find double factorials (n!!), factorials of decimals and negatives with the gamma function, subfactorials, falling and rising factorials, and see how close Stirling's shortcut gets.
Whole numbers give an exact answer (up to 50,000!). Decimals like 4.5 and negatives like −0.5 use the gamma function. 1,000 and 1e6 work too.
x can be negative or a decimal. Falling with x = 10 and k = 3 is 10 × 9 × 8, the same as 10P3.
How your factorial is worked out
Your number: multiply every whole number down to 1.
- 1
Line them up
10 down to 110 numbers10 × 9 × … × 1
- 2
Multiply
10 × 9 = 90 …= 3,628,800
- 3
Count digits and 0s
7 digits2 zeros at the endone per 5 in it
- 4
Check with Stirling
√(2πn) × (n/e)ⁿ≈ 3.599 × 10⁶0.83% low
Read more: the formulas, worked examples, a table and notes
A factorial multiplies a whole number by every whole number below it, down to 1. It counts the ways to put things in order: the first spot has n choices, the next has n − 1, and so on. Factorials grow faster than any power: 10! is over 3.6 million, 20! is over 2.4 billion billion, and 70! is bigger than a googol (10100).
The formula
n! = n × (n − 1) × (n − 2) × … × 2 × 1, and 0! = 1
Zeros at the end of n! = ⌊n ÷ 5⌋ + ⌊n ÷ 25⌋ + ⌊n ÷ 125⌋ + … (⌊ ⌋ means round down)
Double factorial: n!! = n × (n − 2) × (n − 4) × … down to 1 or 2; step k: n × (n − k) × (n − 2k) × …
Decimals and negatives: x! = Γ(x + 1), the gamma function
Subfactorial: !n = n × !(n − 1) + (−1)n, with !0 = 1 and !1 = 0; also n! ÷ e rounded
Falling: x × (x − 1) × … × (x − k + 1). Rising: x × (x + 1) × … × (x + k − 1)
Stirling's approximation: n! ≈ √(2πn) × (n ÷ e)n, closer still × (1 + 1 ÷ (12n))
Worked example
10! (the default): 10 × 9 = 90, × 8 = 720, × 7 = 5,040, × 6 = 30,240, × 5 = 151,200, × 4 = 604,800, × 3 = 1,814,400, × 2 = 3,628,800 (and × 1 changes nothing). It has 7 digits. Zeros at the end: ⌊10 ÷ 5⌋ = 2, and 25 is bigger than 10, so there are 2 zeros (from 10 = 2 × 5 and from 5 × 2).
Stirling check: √(2π × 10) × (10 ÷ 2.71828…)10 ≈ 7.9267 × 453,999.3 ≈ 3,598,696, which is 0.83% below 3,628,800. Times the correction 1 + 1/120 it is about 3,628,685, only 0.0032% low.
Double factorial: 9!! = 9 × 7 × 5 × 3 × 1 = 945 and 10!! = 10 × 8 × 6 × 4 × 2 = 3,840. Check: 3,840 × 945 = 3,628,800 = 10!.
Gamma: 4.5! = Γ(5.5) = (945 ÷ 32) × √π ≈ 29.53125 × 1.7724539 ≈ 52.3428.
Subfactorial: !4 = 4 × !3 + 1 = 4 × 2 + 1 = 9. Four friends in a Secret Santa can draw names in 4! = 24 ways, and in 9 of them nobody gets their own name: 9 ÷ 24 = 37.5%.
Factorial table
| n | n! | n | n! |
|---|---|---|---|
| 0 | 1 | 8 | 40,320 |
| 1 | 1 | 9 | 362,880 |
| 2 | 2 | 10 | 3,628,800 |
| 3 | 6 | 11 | 39,916,800 |
| 4 | 24 | 12 | 479,001,600 |
| 5 | 120 | 13 | 6,227,020,800 |
| 6 | 720 | 15 | 1,307,674,368,000 |
| 7 | 5,040 | 20 | 2,432,902,008,176,640,000 |
Notes
- Whole-number answers are exact up to 50,000! (213,237 digits). Beyond that, up to 1,000,000,000!, the answer comes from Stirling's series and only the figures that can be trusted are shown.
- Gamma values (decimals and negatives) are worked out numerically with the Lanczos method, good to about 15 significant figures, and shown to 10.
- The biggest factorial an ordinary calculator or spreadsheet can show is 170! (about 7.26 × 10306); 171! is too big for normal decimal numbers.
- Where factorials appear: arranging things (n!), choosing groups (nCr = n! ÷ (r! (n − r)!)), probability, the number e = 1/0! + 1/1! + 1/2! + …, and Taylor series in calculus.
Common questions
What is a factorial?
The factorial of a whole number n, written n!, is every whole number from n down to 1 multiplied together. So 5! = 5 × 4 × 3 × 2 × 1 = 120 and 10! = 3,628,800. It counts the ways to put n different things in order: 5 books can stand on a shelf in 120 different orders.
Why is 0! equal to 1?
There is exactly one way to arrange nothing: leave it empty. The rule n! = n × (n − 1)! also needs it, because 1! = 1 × 0!, so 0! must be 1. That is why formulas like nCr = n! ÷ (r! × (n − r)!) still work when r is 0 or n.
How many zeros are at the end of 100!?
Each zero at the end comes from a 10, which needs a 2 and a 5. There are always more 2s than 5s, so count the 5s: 100 ÷ 5 = 20 numbers have a 5, and 100 ÷ 25 = 4 of them have a second one. 20 + 4 = 24 zeros. 100! has 158 digits in all.
What is a double factorial (n!!)?
A double factorial skips every other number: n!! = n × (n − 2) × (n − 4) × … down to 1 or 2. So 9!! = 9 × 7 × 5 × 3 × 1 = 945 and 10!! = 10 × 8 × 6 × 4 × 2 = 3,840. It is not the factorial of a factorial. Multiplying 10!! by 9!! gives 10!.
Can you take the factorial of a decimal or a negative number?
Not by multiplying, but the gamma function extends it: x! = Γ(x + 1). So 0.5! = √π ÷ 2 ≈ 0.8862 and 4.5! ≈ 52.34. Negative whole numbers such as −1 or −3 have no factorial, because the gamma function shoots off to infinity there. Negative halves are fine: (−0.5)! = √π ≈ 1.7725.
What is a subfactorial (!n)?
The subfactorial !n counts the ways to mix up n things so that none ends up in its own place, such as a Secret Santa where nobody draws their own name. !4 = 9 and !10 = 1,334,961. It is n! ÷ e rounded to a whole number, so the chance of a full mix-up is about 1/e, or 36.79%.
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