Hello all! In this article, we’ll talk about the factorial operation, another mathematical operation involving repeated multiplication, but very different from exponentiation.
They are a very useful operation in things like cryptography, probability, and many others!
Definition: What is a Factorial?
The notation for a factorial is n!
n is any integer / whole number.
Although n can be a negative number, that requires a very difficult concept called analytic continuation which we will not delve into for the purposes of this article, as I’m trying to get the very basics figured out.
So, for this article, consider n valid only for whole numbers greater than or equal to 0.
A factorial is a shorthand for multiplication from any whole number down to one.
For example:
- 4! = 4 × 3 × 2 × 1 = 24
- 5! = 5 × 4 × 3 × 2 × 1 = 120
- 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
What Are the Properties of Factorials?
Factorials have a few properties.
- n needs to be a positive integer — as mentioned before this can get more complex but let’s not do that.
- The factorial of a number n must be equal to the product of n times the factorial of the previous integer (e.g. 5! = 5 × 4! = 5 × 4 × 3 × 2 × 1)
- 1! = 1
- The factorial of any integer above 1 will always be greater than or equal to the number itself (e.g. 2! = 2, 3! = 6 et cetera)
- 0! equals, by convention, 1
Why Factorials Matter
Factorials are used in all sorts of ways, from calculating the number of possible arrangements of people at a dinner table to calculating probabilities.
Applications of Factorials in Real Life
How many ways can you arrange 4 people in a line? 4!
No, really, 4! = 4 × 3 × 2 × 1 = 24 possible combinations!
Factorials, as mentioned above, are a shorthand for repeated multiplication of integers from one to n.
They can be used in a lot of ways, particularly interesting to me being the field of cryptography and combinatorics.
When watching a movie and seeing a little keypad with 10 numbers, just remember that the number of combinations the actor must try, if they didn’t have the code already, would be 10! — assuming none of the numbers repeat.
In case the passkey doesn’t use all 10 digits, you’ll have to skip the normal factorial notation and just boringly multiply until you reach the number of possible combinations. For example, if the password uses 5 digits and you know they don’t repeat then the number of combinations is 10 × 9 × 8 × 7 × 6 = 30,240.
You can also see factorials in action when unlocking your phone with a pattern. Most phones have a 3x3 grid, and for those that let you connect two far away points without also connecting the intermediary points, the number of possibilities there is equal to 9! = 362,880.
Some Less Used Things About Factorials
As mentioned above, 0! = 1 by convention. This convention is called the Empty Product Convention. You can learn more about it by visiting the Further Reading section.
As you may know, mathematicians love simplifications — because otherwise, they would lose their brain to stress even faster. As such, a man named James Stirling created something we now call Stirling’s Approximation in his honor.
This is an approximation of all factorial values which has stood the test of time with great accuracy for both small and large values of n.
Fun Facts About Factorials
Did you know that factorial functions generally grow faster than exponential functions? It’s true!
Although the exponential starts off faster, it is quickly beaten by the factorial.
In the example below, the exponential function is x² (dotted blue curve) and the factorial function is x! (solid green curve).
Also, did you know that the number of possible arrangements of a deck of cards can be represented by 52!
That number is larger than the number of seconds that have elapsed since the universe began!
Here Are Some Practice Problems
If you want to learn, here are a few problems that can solidify your built-up understanding of factorials.
- 5! — 3! = ?
- How many ways can you arrange 5 books in a row?
- (6–2)! = ?
- If you need to select 3 out of 6 items, how many ways can you do this? (Hint: This involves combinations and factorials.)
- 7! + 5 = ?
Conclusion
To wrap things up, let’s quickly recap what we learned today:
- Factorials are widely used in fields such as cryptography, combinatorics and probability.
- They are a shorthand for multiplying all integers from n down to 1.
I hope that you have enjoyed today’s article about factorials. If you did, don’t forget to comment what you would like to see me cover next, I’ll make sure to respond as soon as I can!
In the meantime, consider trying to solve the problems above or maybe make your own.
Further Reading
- The example of 52! was inspired by a segment in Vsauce’s Math Magic.
- My article on exponents and logarithms for a quick refresher
- The subreddit r/unexpectedfactorial is a fun read about people unintentionally writing a factorial by following a number with an exclamation mark
- The Wikipedia page for Factorials
- The Wikipedia page for Empty Products
- The Wikipedia page on Stirling’s Approximation
- “Can a negative number have a factorial?”, discussion on the ExplainLikeIm5 subreddit
- The Organic Chemistry Tutor’s video Factorials Explained!
Acknowledgements
- All photos are screenshots created by the author within Symbolab and the Desmos Graphing Calculator app.
Some AI assistance was used to streamline the structure and clarify technical language.
