# Double Factorials and Multifactorials

Before reviewing this article, the readers are expected to know what factorials are first.

The **double factorial** of a positive integer \(n\) is the generalization of the factorial \(n!;\) this type of factorial is denoted by \(n!!\). It is a type of **multifactorial** which will be discussed later. As far as double factorial is concerned, it ends with \(2\) for an even number, and \(1\) for an odd number:

- for an
*even*number \(n>0\), \(n!!=n\times (n-2)\times \cdots\times 4\times 2;\) - for an
*odd*number \(n>0\), \(n!!=n\times (n-2)\times \cdots\times 3\times 1;\) - if \(n=0\), then \(0!!=1.\)

To clarify, \(n!! \) is not equal to \((n!)! \). For example, \(4!! = 4\times2=8,\) whereas \((4!)! = 24! = 620448401733239439360000\). Go ahead and try out the following warm-up problem:

## Some Useful Theorems

For any non-negative integer \(n,\) we find that

\[\dfrac{n!}{n!!}=(n-1)!! ~~\text{ or }~~ n!=(n-1)!!×n!!.\]

We have the following 2 cases:

- If \(n\) is
odd, \[\dfrac{n!}{n!!}=\dfrac{n\times (n-1)\times (n-2)\times \cdots \times 3\times 2\times 1}{n\times (n-2)\times (n-4)\times \cdots 5\times 3\times 1}.\] Since all the odd numbers \(n, n-2, n-4, \ldots , 5, 3\) get canceled, we are left with the equation \[\dfrac{n!}{n!!}=(n-1)!!.\]- If \(n\) is
even, \[\dfrac{n!}{n!!}=\dfrac{n\times (n-1)\times (n-2)\times \cdots \times 3\times 2\times 1}{n\times (n-2)\times (n-4)\times \cdots \times 4\times 2}.\] Since all the even numbers \(n, n-2, n-4, \ldots , 4, 2\) get canceled, we are left with the equation \[\dfrac{n!}{n!!}=(n-1)!!.\]Combining both cases, we find that for

anynon-negative integer \(n\), \[\dfrac{n!}{n!!}=(n-1)!!. \ _\square\]

Suppose that \(n!!\) is defined as follows:

\[ n!! = \begin{cases} n \times (n-2) \times \cdots \times 5 \times 3 \times 1 &\text{if } n \text{ is odd}; \\ n \times (n-2) \times \cdots \times 6 \times 4 \times 2 &\text{if } n \text{ is even}; \\ 1 &\text{if } n = 0, - 1. \\ \end{cases} \]

Then what is

\[\color{red}{\dfrac{9!}{6!!}} \div \color{green}{\dfrac{9!!}{6!}}?\]

For any non-negative integer \(n,\) we find that

\[\dfrac{(2n+1)!}{(2n)!!}=(2n+1)!!.\]

Here, there is no need to consider two separate cases because it makes no difference whether \(n\) is odd or even.

We can expand the LHS as

\[\dfrac{(2n+1)\times (2n)\times (2n-1)\times \cdots \times 3\times 2\times 1}{(2n)\times (2n-2)\times (2n-4)\times \cdots \times 4\times 2}.\]

Since all the even numbers \(2n, 2n-2, 2n-4, \ldots, 4, 2\) get canceled, we are left with the equation

\[\dfrac{(2n+1)!}{(2n)!!}=(2n+1)!!. \ _\square\]

Evaluate \(\frac {9!}{9!!}\).

Since \(\frac {n!}{n!!}=(n-1)!!\), substituting the values, we get

\[\begin{align} \dfrac{9!}{9!!}&=(9-1)!!\\ &=8!!\\ &=8×6×4×2\\ &=384. \ _\square \end{align}\]

Evaluate \(\frac {(3!)!}{3!!}\).

We have

\[\begin{align} \dfrac {(3!)!}{3!!} &=\dfrac {(3×2×1)!}{3×1}\\ &=\dfrac {6!}{3}\\ &=\dfrac {6×5×4×3×2×1}{3}\\ &=\dfrac {720}{3}\\ &=240. \ _\square \end{align}\]

\[\Large{\color{green}{\dfrac{9!}{8!!}}} \div {\color{orange}{\dfrac{7!}{6!!}}} = \, ? \]

**Notation:**

\[ n!! = \begin{cases} n \times (n-2) \times \cdots \times 5 \times 3 \times 1 && \text{if } n \text{ is odd;} \\ n \times (n-2) \times \cdots \times 6 \times 4 \times 2 && \text{if } n \text{ is even;} \\ 1 && \text{if } n = 0, - 1. \\ \end{cases} \]

##### Try the first part here!

For any non-negative integer \(n\) we find that

\[\dfrac{(2n-1)!}{(2n-2)!!}=(2n-1)!!.\]

Again here, there is no need to consider two separate cases. We can expand the LHS as

\[\dfrac{(2n-1)\times (2n-2)\times (2n-3)\times \cdots \times 3\times 2\times 1}{(2n-2)\times (2n-4)\times \cdots \times 4\times 2}.\]

Since all the even numbers \(2n-2, 2n-4, \ldots , 4, 2\) get canceled, we are left with the equation

\[\dfrac{(2n-1)!}{(2n-2)!!}=(2n-1)!!. \ _\square\]

Evaluate \(\frac{9!}{8!!}\).

We have

\[ \begin{align}

\dfrac { 9! } { 8!!} &= \dfrac{ (2 \times 5 - 1 ) ! } { (2 \times 5 - 2 ) !! } \\ &= (2 \times 5 - 1 )!! \\ &= 9 !! \\ &= 9 \times 7 \times 5 \times 3 \times 1 \\ &= 945. \ _\square \end{align} \]

**Cite as:**Double Factorials and Multifactorials.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/double-factorials-and-multifactorials/