Factorial

Bill Gosper was very proud about his discovery, though it is hard to tell by looking at him.

In mathematics, factorial is a unary operator on the set of natural numbers. Go give n any numbers, but n can't be less than 0. The number must not be as big as Hell, as Rayo's number is. The "n factorial", denoted by $n!$, is the product of the first n positive natural numbers, viz:

$$n!=1\times 2\times 3\times 4\times \dots \times n$$

To get it through the thicker of skulls, we could also write it as:

$$n!=n\times (n-1)\times (n-2)\times \dots \times 3\times 2\times 1$$

In particular, n is not defined when it is a negative integer. Also, if n = 0, the rules of baseball declare that $0!=1$, following the empty product rule. Or we can propose against the rule that it is undefined by using the formula $n!=n(n-1)!$. Look! Someone else is churning the mathematical law by proving that it is 0 lmao.

$$0!=0\times (0-1)!=0\times (-1)!=0$$

The factorial is popular in many different areas of mathematics and is also helpful in geometry, and most notably combinatorics, since it is the number of different ways to shuffle a group of some n objects. In combination, the formula is: ${_{n}}C_{k}={\frac {n!}{k!(n-k)!}}$ (with $0<k\leq n$ condition)

Representation

"n factorial" is written n! Charles Schulz popularized this notation in his comic strip Peanuts. Lead character Charlie Brown heard barn swallows chirping every morning (the chirps represented in the comic as "!", and he stupidly assumed they must be discussing the properties of the factorial function. When the birds got especially agitated, Schulz wrote "!!" — a representation that naturally matches the double factorial (see below).

Stirling approximation

An approximation of the factorial is useful for real and gigantic values of n. This worthy formula gives accurate results even for small n, but is utterly useless if n dips below zero. It is named after James Stirling, although it is Moivre's discovery and Stirling probably stole it.

$$n!\sim {\sqrt {2\pi n}}\cdot e^{-n}n^{n}$$

When the formula's accuracy is too low, we have the Gosper's approximation:

$$n!\sim {\sqrt {\left(2n+{\frac {1}{3}}\right)\pi }}\,n^{n}e^{-n}$$

Or this one, discovered by an Indian physician named Ramanujan:

$$n!\approx {\sqrt {\pi }}\left({\frac {n}{e}}\right)^{n}{\sqrt[{6}]{{8n^{3}}+{4n^{2}}+n+{\frac {1}{30}}}}$$

Double factorial

We can consider the $n!!$ as the product of the first n elements of the arithmetic progression with the first element equal to 1 and the common difference equal to 1. Expanding with the common difference equal to 2 we have:

$$n!!=\left\{{\begin{matrix}1\qquad \qquad \ &&\\n(n-2)!!&&\qquad \qquad \end{matrix}}\right.$$

It is the product of the first n elements of the sequence with the first element 1 and the common difference is 2. For even integers, the formula is:

$$(2n)!!=\prod _{k=1}^{n}2k$$

Double factorial of egg for example:

$$8!!=\prod _{k=1}^{4}2k=384$$

Not like the dumb factorial, double factorial has a valid value for some odd negative integers, but not even negative integers. For even negative integers, it is undefined. An overview about the special formula of double factorial:

$(n-2)!!={\frac {n!!}{n}}$