Factorial

The factorial of a positive integer n equals the product of all positive integers less than or equal to n. This is denoted as n!

Mathematically, the formal expression for factorial is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n!=\prod_{k=1}^n k \qquad \mbox{for all } n \in \mathbb{N} \ge 0. \!}

For example,

$4!=1\cdot 2\cdot 3\cdot 4=24\$

In the case of 0, sometimes a special definition declares that

$0!=1\$

even though one would expect that to be zero. This special definition is justified by saying that the product of no numbers is equal to 1. Another way to justify it is to observe that in general,

$(n-1)!={\frac {n!}{n}}$

so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0! = (1-1)! = \frac{1!}{1} = \frac{1}{1} = 1}

The number 7! in Plato's Laws

A curious use of factorials occurs in Plato's Laws. In Book V he says the citizens of his ideal state should be exhorted:

"Good friends, honor order and equality, and above all the number 5040.

5040 is 7! or seven factorial,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7 ! = 1\cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7= 5040 \ }

The ideal state is to contain exactly 5040 citizens. His reason is that this number can be divided exactly by every number from 1 to 10, as well as 12, 20, 21, and 144, which he seems to think is important:

We will fix the number of citizens at 5040, to which the number of houses and portions of land shall correspond. Let the number be divided into two parts and then into three; for it is very convenient for the purposes of distribution, and is capable of fifty-nine divisions, ten of which proceed without interval from one to ten. Here are numbers enough for war and peace, and for all contracts and dealings. These properties of numbers are true, and should be ascertained with a view to use.