The factorial of a natural number $n$ is the number of permutations of $n$ elements, or equivalently the number of bijections from an $n$-element set to itself. It is denoted $n!$. For example, $7! = 5040$. But where does the notation come from?
In 1827, Thomas Jarrett tried to introduce the notation
for the factorial of $n$. This notation has some problems. First, it is inconvenient to draw a line under the character. Also, if you are writing it by hand, then you have to move your pencil backwards, assuming you start at the top of the vertical line, which is how most people would probably draw it.
Luckily, it was Christian Kamp who introduced a better notation, the exclamation point $n!$ that we use today. He introduced it in his book Élémens d’arithmétique universelle a little earlier than Jarrett in 1808. He wrote:
Je me sers de la notation trés simple $n!$ pour désigner le produit de nombres décroissans depuis n jusqu’à l’unité, savoir $n(n-1)(n-2) … 3.2.1$. L’emploi continuel de l’analyse combinatoire que je fais dans la plupart de mes démonstrations, a rendu cette notation indispensable.
Here is my translation:
I am using the very simple notation $n!$ to designate the product of numbers decreasing from $n$ to $1$, namely $n(n-1)(n-2) … 3.2.1$. The constant use of combinatorial analysis in most of my demonstrations makes this notation indispensable.
This notation still has some oddities. If factorials appear at the end of a sentence, sometimes they look unusual, like 5!. Also, if you want to include an exclamation point like: I love 5!! In this case there could be some ambiguity between the regular factorial and the double factorial, which is another function altogether. Good thing mathematicians don’t make much use of exclamation marks in writing.
In any case, the factorial notation is here to stay. Personally, I think it is pretty good.