# How to Use the Factorial Notation

We have had several discussions about the factorial notation, so I think this introduction is a bit late. However, it is important that you know these basic facts in order to perform calculations and understand better in later discussions.

The factorial of a non-negative integer $n$ is the product of all the positive numbers less than $n$. For example, the

$4! = 4 \times 3 \times 2 \times 1 = 24$.

and

$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.

In Introduction to Permutations, we have discussed that there are $n!$ ways to arrange $n$ distinct objects into a sequence. For instance, if we have 3 objects namely A, B, and C, then they can be arranged in $3! = 3 \times 2 \times 1 = 6$ ways. The arrangement are as follows:

$ABC, ACB, BAC, BCA, CAB, CBA$.

We have also learned some reasons why mathematicians chose the definition $0! = 1$.

Some Factorial Facts and Fun

The definition above translates that

$n! = n(n-1)(n-2) \cdots (3)(2)(1)$

or the product of the non-negative integer $n$ and all the positive integers below it. From this definition, it is easy to see the following basic facts.

1.) $n! = n(n - 1)!$

because $n! = n(n-1)(n-2) \cdots (3)(2)(1)$ and $(n-1)! = (n - 1)(n-2) \cdots (3)(2)(1)$.

2.) $\frac{n!}{(n-1)!} = n$ which directly follows from (1)

3.) A prime number that is 1 less or 1 more than a value of a factorial is called a factorial prime. Therefore, 7 is a factorial prime since $7 = 3! + 1$.

4.) Factorials are also used in Calculus particularly in Taylor’s Theorem.

5.) You can use factorials to remember time!

There are 4! hours in a day.

There are 8! minutes in four weeks.

There are 10! seconds in six weeks.

Below is a video about the basics of factorial from Numberphile including some explanations of why ! = 1 which is also discussed in this post.

Enjoy learning!