Factorial

Elementary combinatorics: Factorial and binomial coefficient

Factorial

Just as you shorten $\blue{\overbrace{\color{black}{a \times a \times \cdots \times a}}^{n\mathrm{\;times}}}$ to $a^n$ , you also have a short notation for $n\cdot (n-1)\cdots 3\cdot 2\cdot 1$ , which is $n!$ . We pronounce this as `` $n$ factorial''.

The factorial of a natural number $n$ is defined as follows:

$\begin{cases} n! =n\cdot (n-1)\cdots 3\cdot 2\cdot 1 & \text{for } n\in\mathbb{N}\backslash\{0\}\\ 0!=1&\end{cases}$

We can also define the factorial via an improper integral in the form of the gamma function:

$\Gamma(x)=\int_{0}^{\infty} t^{x-1}e^{-t}\,\dd t$ For natural numbers $n$ then holds:

$n!=\Gamma(n+1)$ But because the gamma function exists for all numbers except negative integers, you can extend the factorial to more than only the natural numbers. For example:

$\begin{aligned} (-\tfrac{1}{2})! &= \Gamma(\tfrac{1}{2})\\[0.25cm] &= \int_{0}^{\infty} t^{-\frac{1}{2}}e^{-t}\,\dd t\\[0.25cm] &=2\cdot \int_{0}^{\infty} e^{-u^2}\,\dd u & \blue{\text{substitution }u=t^{\frac{1}{2}}\text{ and } \dd u=t^{-\frac{1}{2}}\,\dd t}\\[0.25cm] &= 2\cdot \frac{\sqrt{\pi}}{2} & \blue{\text{standard integraa}}\\[0.25cm] &= \sqrt{\pi}\end{aligned}$ The graph of the factorial function looks as follows:

grafiek van de faculteitsfunctie

The number of ways in which $n$ different objects can be arranged is equal to $n!$ .

In other words, the number of permutations of $n$ things (i.e. the number of arrangements of $n$ things) is equal to $n!$ .

The number of permutations of $n$ things of which $m$ are the same and the remainder is different is equal to $\dfrac{n!}{m!}$ .

For all $m,n\in\mathbb{N}$ with $m<n$ the following rules apply:

$\begin{aligned} n!&=n\cdot (n-1)!\\[0.25cm] \frac{n!}{m!} &= (m+1)\cdot (m+2)\cdots (n-1)\cdot n\end{aligned}$