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 is the product of all the positive numbers less than . For example, the
In Introduction to Permutations, we have discussed that there are ways to arrange distinct objects into a sequence. For instance, if we have 3 objects namely A, B, and C, then they can be arranged in ways. The arrangement are as follows:
We have also learned some reasons why mathematicians chose the definition .
Some Factorial Facts and Fun
The definition above translates that
or the product of the non-negative integer and all the positive integers below it. From this definition, it is easy to see the following basic facts.
because and .
2.) 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 .
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.