Factorial

The factorial of a number n is equal to the product of all positive integers that are less than or equal to n. We write the factorial using an exclamation mark: n!. For zero, 0! = 1. The factorial is used mainly in combinatorics, where it is needed to calculate, for example, permutations. For example, the factorial of five is 5! = 5 · 4 · 3 · 2 · 1 = 120.

Definition of the factorial

We can define the factorial in various ways. The simplest is probably this one:

$$n!=\{\begin{array}{ll}1& \text{if }n=0\\ n\cdot (n-1)!& \text{otherwise}\end{array}$$

We can also give a definition using a product:

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

Or using an integral:

$$(z-1)!=\mathrm{\Gamma }(z):={\int }_0^{\mathrm{\infty }}{t}^{z-1}{\mathrm{e}}^{-t}\mathrm{d}t,\mathrm{\Re }(z)>0.$$

Calculating with factorials

Factorials are often used in fractions. Here we use the fact that n! = n · (n − 1)!. This holds by definition, and we’ll show it with an example. We know that the factorial of four, 4!, is equal to the product of these positive integers: 4 · 3 · 2 · 1. We can write that the product 3 · 2 · 1 is equal to the factorial of three: 3 · 2 · 1 = 3!. The factorial of four can then be written as 4! = 4 · 3!.

$$4!=4\cdot \underset{=3!}{\underbrace{3\cdot 2\cdot 1}}=4\cdot 3!$$

Thanks to this, we can effectively cancel many different factorials in fractions. A sample problem – simplify the expression:

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

We simplify it using the formula I mentioned a moment ago. In the numerator we can split the factorial into n · (n − 1)!, and this can be broken down further (using the same formula) into n · (n − 1) · (n − 2)!. Now we can cancel the fraction nicely:

$$\frac{n!}{(n-2)!}=\frac{n(n-1)\boxed{\text{(n-2)!}}}{\boxed{\text{(n-2)!}}}=n(n-1)$$

Another example:

$$\begin{array}{rcl}\frac{n!\cdot (n+1)!}{(n-1)!\cdot (n+2)!}& =& \frac{n\cdot \boxed{\text{(n-1)!}}\cdot (n+1)!}{\boxed{\text{(n-1)!}}\cdot (n+2)!}\\ & =& \frac{n\cdot (n+1)!}{(n+2)!}\\ & =& \frac{n\cdot \boxed{\text{(n+1)!}}}{(n+2)\cdot \boxed{\text{(n+1)!}}}\\ & =& \frac{n}{n+2}\end{array}$$

And the last example with factorials:

$$\begin{array}{rcl}\frac{(2(n+1))!}{(2n)!}& =& \frac{(2n+2)!}{(2n)!}\\ & =& \frac{(2n+2)(2n+1)\boxed{\text{(2n)!}}}{\boxed{\text{(2n)!}}}\\ & =& (2n+2)(2n+1)\end{array}$$