What is a “factorial,” and how is it used in mathematics?

## The Mathematical Definition of Factorial for Positive Integers

In mathematics, a factorial is a function applied to natural numbers greater than zero. The symbol for the factorial function is an exclamation mark after a number, like this: 2!

The usual definition of “n factorial” is “n! = n * (n – 1) * (n – 2) *…* 2 * 1,” where ‘n’ is a positive integer.

The first handful of factorial values from positive integers are: 1! = 1; 2! = 2; 3! = 3*2 = 6; 4! = 4*3*2 = 24; and 5! = 5*4*3*2 = 120. In the image, only the first five factorial results are plotted although the first ten have been calculated.

## Recursive Definition for the Factorial

A recursive definition for the factorial function is “n! = n * (n-1)!”, placing the lower limit for the recursion at n=2.

## Factorial Function: Practical Uses

Although the factorial function deals with repeated multiplication, its most obvious use in math is to compute the *number of ways in which ‘n’ objects can be permuted*.

A permutation is a re-arrangement of a set. For example, set A={a} has exactly one arrangement. However, set B={a,b} could be re-arranged as {b,a}; and set C={a,b,c} has the six permutations {(a,b,c), (a,c,b), (b,a,c), (b,c,a), (c,a,b), (c,b,a)}.

Note that |A| = 1 (set A has one element), |B| = 2 and |C| has three; but they have 1, 2 and 6 permutations respectively. Our readers are invited to list the permutations for set D={a,b,c,d}, but there should be two dozen in that permutation set.

This extends to selecting permutations of *an ordered subset*. From set C={a,b,c}, how many ways can two elements be selected and permuted? The solution set is {(a,b), (b,a), (a,c), (c,a), (b,c), (c,b)}, and has six elements.

**Click to Read Page Two: The Formula for Choosing K**

Lee Shafer says

I just saw the clock where all the hours are expressed as some formula involving only 9. I don’t get 7 o’clock. To make 7 you need that last part to =1. There is a line with .9 under it. I never had that in school.what is that?it obviously equals one. Lee

Mike DeHaan says

Excellent question. By appeal to authority in Wolfram, “The special case 0! is defined to have value 0!=1, consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set)”.

Wikipedia’s “Gamma function” has the line

“Combining this with Γ(1) = 1, we get Γ(n) = (n-1)!”

when defining the gamma function.

Plus…

SosMath discusses the gamma function as an integral, noting that “e^(-1) = 1″ (and should have noted “x^0 = 1″). Further down the page, they derive Γ(1) = 1 from the integral.

(See http://sosmath.com/calculus/improper/gamma/gamma.html ).

My basic answer is: it works out nicely when relating factorials to the gamma function.

(Sorry I hadn’t checked my mail over the weekend!)

joe franco says

why is 0! = 1

meshfields_blog says

Because the number of possible permutations of 0! is 1. The empty set { } is still one empty set with one permutation inside.

Proof: Let n=1, using the definition n! = n*(n-1)! then is

1! = 1*0! which can be simplified to 1 = 0!. ■