Multifactorial — Definition, Formula & Examples
A multifactorial is a generalization of the factorial where you multiply a positive integer by every th integer below it, rather than every integer below it. For example, the double factorial multiplies every second integer down from , and the triple factorial multiplies every third integer.
For a positive integer and a positive integer , the -th multifactorial of , denoted or with exclamation marks, is defined as where the product continues as long as each factor is greater than zero.
Key Formula
Where:
- = The starting positive integer
- = The step size (number of integers skipped plus one)
- = Index running from 0 up to $\lfloor (n-1)/k \rfloor$
How It Works
To compute a multifactorial, start with and repeatedly subtract , multiplying each result, until the next subtraction would give zero or a negative number. For a double factorial (), you skip every other integer: . For a triple factorial (), you skip two at a time: . When , you recover the ordinary factorial since you subtract 1 each time.
Worked Example
Problem: Compute the double factorial 9!!.
Identify the pattern: A double factorial means , so multiply every second integer down from 9.
Multiply step by step: Work left to right: , then , then , then .
Answer:
Why It Matters
Double factorials appear in combinatorics when counting perfect matchings and in probability formulas involving the Gaussian distribution. They also show up in physics, particularly in the coefficients of series expansions in quantum mechanics and statistical mechanics.
Common Mistakes
Mistake: Interpreting as — that is, taking the factorial of and then taking the factorial of the result.
Correction: The notation means the double factorial (multiply every second integer down from ), not the factorial applied twice. would be astronomically larger.