Factorial

The factorial of a whole number ${\displaystyle n}$, written as ${\displaystyle n!}$ or ${\displaystyle n}$, is found by multiplying ${\displaystyle n}$ by all the whole numbers less than it. For example, the factorial of 4 is 24, because ${\displaystyle 4\times 3\times 2\times 1=24}$. Hence one can write ${\displaystyle 4!=24}$. For some technical reasons, 0! is equal to 1.

Factorials can be used to find out how many possible ways there are to arrange ${\displaystyle n}$ objects.

For example, if there are 3 letters (A, B, and C), they can be arranged as ABC, ACB, BAC, BCA, CAB, and CBA. That is be 6 choices because A can be put in 3 different places, B has 2 choices left after A is placed, and C has only one choice left after A and B are placed. In other words, ${\displaystyle 3\times 2\times 1=6}$ choices.

The factorial function is a good example of recursion (doing things over and over), as ${\displaystyle 3!}$ can be written as ${\displaystyle 3\times 2!}$, which can be written as ${\displaystyle 3\times 2\times 1!}$ and finally as ${\displaystyle 3\times 2\times 1\times 0!}$. Because of this, ${\displaystyle n!}$ can also be defined as ${\displaystyle n\times (n-1)!}$, with ${\displaystyle 0!=1}$

The factorial function grows very fast. There are ${\displaystyle 10!=3,628,800}$ ways to arrange 10 items.