Multifactorial Calculator (n!!, n!!!)
Calculate the multifactorial n!!, n!!! or any step-k factorial, with the full multiplication chain shown. Clears up the classic n!! vs (n!)! mix-up.
How to use this calculator
Enter n and the step (k) — k = 1 gives the ordinary factorial n!, k = 2 gives the double factorial n!!, k = 3 gives n!!!, and so on up to k = 9. The calculator shows the live notation, multiplies n downward in steps of k, and gives the exact value.
What is a multifactorial?
The ordinary factorial n! multiplies every integer from n down to 1. A multifactorial instead skips ahead by a fixed step k each time:
| Notation | Step | Example |
|---|---|---|
| n! | 1 | |
| n!! | 2 | |
| n!!! | 3 |
More generally, n with k exclamation marks multiplies n, n−k, n−2k, … stopping once the next term would drop to 0 or below.
The classic mix-up: n!! is NOT (n!)!
This is the single most common source of errors with multifactorial notation.
- (double factorial) — step straight down by 2: .
- (factorial of a factorial) — compute first, then take its factorial: , a number with more than a million digits.
The two are wildly different in size for anything beyond the smallest inputs.
Whenever you see n!! in a formula, read it as "double factorial," not
"factorial applied twice."
Even n vs. odd n
Because the double factorial steps down by 2, the parity of n determines which numbers appear in the chain:
- Even n lands on 2: .
- Odd n lands on 1: .
Both follow the exact same rule (subtract 2 each step); they simply end at a different last term depending on whether n is even or odd.
Where multifactorials show up
- Combinatorics. Double factorials count certain pairings and perfect matchings — for instance, the number of ways to pair up 2n objects is .
- Geometry. Formulas for the volume and surface area of n-dimensional spheres involve double factorials.
- Probability. Expansions of the Gaussian (normal distribution) integral produce double-factorial terms.
Worked example — 9!! and 9!!!
Double factorial ():
- Start at 9, step down by 2: 9, 7, 5, 3, 1.
So 9!! = 945.
Triple factorial ():
- Start at 9, step down by 3: 9, 6, 3.
So 9!!! = 162.
Frequently asked questions
- What does n!! mean?
- n!! is the *double factorial*: multiply n by every second integer below it down to 1 or 2. For example and . It is not the same as applying the ordinary factorial twice.
- Is n!! the same as (n!)!?
- No — this is the single most common mix-up with this notation. means take first, then take the factorial of *that* number, which explodes to an astronomically larger value. (double factorial) instead multiplies straight down in steps of 2. For : , while , a number with well over a million digits.
- What is a triple factorial (n!!!)?
- Same idea, one more step: multiplies n downward in steps of 3. , and . In general, k exclamation marks means step down by k each time.
- Why do even and odd n give different-looking chains?
- For the double factorial, an even n keeps multiplying even numbers down to 2 (e.g. ), while an odd n multiplies odd numbers down to 1 (e.g. ). Both chains follow the same rule — step down by 2 — they just land on different parity.
- What is 0!! or 0!!!?
- By the same empty-product convention as 0! = 1, any 0 with any number of exclamation marks equals 1 — there is nothing to multiply, so the product is defined as 1.
- Where is the double factorial used?
- It shows up in combinatorics (counting certain pairings/matchings), in the volume formulas for n-dimensional spheres, and in probability (Gaussian integral expansions). It is a genuinely different function from the ordinary factorial, not just a stylistic variant.