A multifactorial is a factorial that skips numbers. Instead of multiplying n by every smaller whole number, it multiplies by every second one (the double factorial n!!), every third one (the triple factorial n!!!) or, in general, every k-th one. The extra exclamation marks do not mean “take the factorial again”. This calculator computes any multifactorial exactly, shows each factor, and checks double factorials against the ordinary factorial.
How to use the multifactorial calculator
- Enter Number (n), a whole number from 0 to 100,000.
- Enter Step size k (number of !): 2 for n!!, 3 for n!!!, 1 for the ordinary factorial, or any step up to 1,000.
- Read the result on the tape, with the number of factors, the smallest factor, the digit count and n! for comparison.
- Use the table under the steps to see the same multifactorial for every number from 0 up to at least 15.
Multifactorial formula
For step size k, keep subtracting k from n and multiply every positive term:
The product stops at the last positive factor, which is between 1 and k. When n is 0 there is nothing to multiply, so the result is 1. The double factorial has two closed forms:
Worked examples
9!! = 9 × 7 × 5 × 3 × 1 = 945. Check: 9! ÷ (24 × 4!) = 362,880 ÷ 384 = 945.
8!! = 8 × 6 × 4 × 2 = 384. Check: 24 × 4! = 16 × 24 = 384.
10!!! = 10 × 7 × 4 × 1 = 280, four factors stopping at 1.
Notice that 9!! × 8!! = 945 × 384 = 362,880 = 9!. The odd and even double factorials split the ordinary factorial between them.
Values for small n
| n | n!! | n!!! | n!!!! |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 2 | 2 | 2 |
| 3 | 3 | 3 | 3 |
| 4 | 8 | 4 | 4 |
| 5 | 15 | 10 | 5 |
| 6 | 48 | 18 | 12 |
| 7 | 105 | 28 | 21 |
| 8 | 384 | 80 | 32 |
| 9 | 945 | 162 | 45 |
| 10 | 3,840 | 280 | 120 |
| 11 | 10,395 | 880 | 231 |
| 12 | 46,080 | 1,944 | 384 |
Where double factorials appear
Pairing people up. Split 2m people into m pairs: the first person has 2m − 1 possible partners, the next unpaired person has 2m − 3, and so on, giving (2m − 1)!! pairings. Six people can be paired 5!! = 15 ways; a class of 20 can be paired 19!! = 654,729,075 ways.
Integrals and geometry. Wallis’s formulas for the integral of sinⁿ x from 0 to π/2 are ratios of double factorials, and so are the volumes of balls in higher dimensions.
Statistics. The even moments of a standard normal distribution are E[Z²ᵐ] = (2m − 1)!!, so E[Z⁴] = 3 and E[Z⁶] = 15.
Common mistakes
- Reading n!! as (n!)!. The double factorial of 4 is 8; the factorial of 4! is 24!, a 24-digit number.
- Forgetting where to stop. 10!! ends at 2, not 1, and 10!!! ends at 1 because 10 − 9 = 1. The calculator reports the smallest factor so you can check.
- Mixing notations. Some books write n!₂ or n!⁽²⁾ for n!!. For steps above 5 this calculator writes the step as a subscript, such as 20!₍₆₎, instead of a long row of exclamation marks.
For the ordinary factorial and its properties — trailing zeros, digit counts and Stirling’s approximation — use the factorial calculator.
Frequently asked questions
What is a double factorial?
The double factorial n!! multiplies n by every second number below it, stopping at 1 or 2. For example, 9!! = 9 × 7 × 5 × 3 × 1 = 945 and 8!! = 8 × 6 × 4 × 2 = 384.
Is n!! the same as (n!)!?
No. n!! skips every other factor, so it is much smaller than n!, while (n!)! is the factorial of a factorial and is enormous. For example, 4!! = 8, but (4!)! = 24! ≈ 6.2 × 10^23.
What are 0!! and 1!!?
Both equal 1. With no positive factors, 0!! is the empty product, matching the convention 0! = 1, and 1!! is simply 1. The same holds for any step size k.
How is the double factorial related to the ordinary factorial?
For even numbers, (2m)!! = 2^m × m!. For odd numbers, (2m + 1)!! = (2m + 1)! ÷ (2^m × m!). Also n! = n!! × (n − 1)!!, because the two double factorials split the factors into odds and evens.
Where are double factorials used?
They count the ways to split 2m people into pairs, which is (2m − 1)!!. They also appear in integrals of powers of sine and cosine, in the volume of higher-dimensional spheres and in the moments of the normal distribution.