Rearranging objects with factorials

We will often encounter situations where we will like to rearrange objects. In fact, we have used the multiplication principle in earlier sections to rearrange objects.

As an example, let us consider four objects $A,B,C$ and $D.$ To rearrange them, we can use the multiplication principle: there are 4 ways to decide which item will be first. Subsequently, there will be 3 ways to decide which item goes next, followed by 2 ways for the next, and 1 way for the last.

Hence the number of ways to rearrange 4 objects is given by $4\times 3\times 2\times 1.$ Since this technique occur so often, we give it a special name and symbol: the factorial, $n!$

Technique: Rearrangements using the factorial, $n!$

There are $n!$ ways to rearrange $n$ unique objects in a line.

The factorial of a positive integer $n,$ denoted $n!,$ is defined by $n!=n(n-1)(n-2)\cdots (1).$

For example, $4!=4(3)(2)(1)=24.$