3,840
3,628,800
945
5
10
2
8.253228
3,840
3,628,800
945
5
10
2
8.253228
The Multifactorial Calculator computes generalized factorials where you skip by a step size $$k$$ instead of multiplying every consecutive integer. The double factorial ($$n!!$$) multiplies every other integer, the triple factorial ($$n!!!$$) multiplies every third integer, and so on.
The formal definition of the $$k$$-th multifactorial is:
$$n!^{(k)} = n \times (n-k) \times (n-2k) \times \cdots$$
where the product continues while terms remain positive. For the double factorial specifically:
$$n!! = \begin{cases} n \times (n-2) \times \cdots \times 2 & \text{if } n \text{ is even} \\ n \times (n-2) \times \cdots \times 1 & \text{if } n \text{ is odd} \end{cases}$$
Multifactorials appear in many areas of advanced mathematics and physics. The double factorial is essential in evaluating Gaussian integrals, computing Wallis-type products, and expressing the volume of hyperspheres. In combinatorics, $$(2n-1)!!$$ counts the number of ways to pair $$2n$$ objects (perfect matchings). The triple factorial appears in certain hypergeometric function identities and series expansions in quantum mechanics.
The calculator also shows the regular factorial $$n!$$ for comparison, letting you see how multifactorials relate to standard factorials through their ratio.
The calculator multiplies all terms of the form $$n - jk$$ that are positive:
$$n!^{(k)} = \prod_{j=0}^{m-1} (n - jk) \quad \text{where } m = \lceil n/k \rceil$$
For k = 1 (standard factorial): $$10!^{(1)} = 10 \times 9 \times 8 \times \cdots \times 1 = 3628800$$
For k = 2 (double factorial): $$10!! = 10 \times 8 \times 6 \times 4 \times 2 = 3840$$
For k = 3 (triple factorial): $$10!!! = 10 \times 7 \times 4 \times 1 = 280$$
The implementation uses up to 15 terms (sufficient for $$n \leq 30$$), with terms that would go below 1 automatically replaced by 1 (multiplicative identity). The term count equals $$\lceil n/k \rceil$$.
The ratio $$n!/n!^{(k)}$$ tells you how many additional factors the standard factorial includes. For $$k=2$$, the double factorial captures roughly the square root of the full factorial in logarithmic terms. A larger step $$k$$ means fewer terms multiplied, yielding a smaller result. The term count shows exactly how many integers participate in the product.
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10!! = 10 × 8 × 6 × 4 × 2 = 3840. Compare to 10! = 3,628,800 — the ratio is 945 = 9!! (the odd double factorial).
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12!!! = 12 × 9 × 6 × 3 = 2160. Only 4 terms are multiplied with step size 3.
The double factorial $$n!!$$ multiplies every other integer from $$n$$ down. For even $$n$$: $$8!! = 8 \times 6 \times 4 \times 2 = 384$$. For odd $$n$$: $$7!! = 7 \times 5 \times 3 \times 1 = 105$$. It is not the factorial of the factorial — that would be $$(n!)!$$, a much larger number.
Double factorials appear in the Wallis product for $$\pi$$, in the volume of n-dimensional spheres ($$V_n \propto \pi^{n/2}/\Gamma(n/2+1)$$), in Gaussian integrals ($$\int_0^\infty x^{2n}e^{-x^2}dx = \frac{(2n-1)!!\sqrt{\pi}}{2^{n+1}}$$), and in the expansion coefficients of Legendre polynomials.
For even $$n = 2m$$: $$n! = n!! \times (n-1)!!$$, meaning the full factorial splits into the product of even and odd double factorials. Also, $$(2m)!! = 2^m \cdot m!$$. For example, $$10!! = 2^5 \cdot 5! = 32 \times 120 = 3840$$.
Yes, via generalizations of the Gamma function. The double factorial extends to complex arguments through $$z!! = 2^{z/2} \pi^{(1-\cos(\pi z))/4} \cdot \Gamma(z/2 + 1) / \Gamma(1/2)^{(1-\cos(\pi z))/2}$$, though the standard integer definition is far more common in practice.
Multifactorials grow slower than regular factorials because they skip terms. The double factorial grows roughly as $$n!! \sim \sqrt{2}(n/2)^{n/2}e^{-n/2}$$, which is much slower than $$n! \sim \sqrt{2\pi n}(n/e)^n$$. The larger the step $$k$$, the slower the growth.
The ratio equals the product of all integers from 1 to $$n$$ that are not included in the multifactorial. For $$n!!/n!! = (n-1)!!$$ when $$n$$ is even. This ratio helps convert between multifactorial and standard factorial expressions in derivations.
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The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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