I understand that there are 4! = 24 ways to arrange 4 objects but what about the title of this post?
Seems to me that its not intuitive and the human brain is not built to understand it.
In: Mathematics
So, normally the factorial function isn’t continuous – it is defined only on non-negative integer values (0,1,2,…).
But what if we wanted to extend this function to all (or most) of the real numbers, and not just integers?
The [Gamma function](https://en.wikipedia.org/wiki/Gamma_function) is one such extension. It is defined in a way that makes it [smooth](https://en.wikipedia.org/wiki/Smoothness) but also matches the factorial on the integers, specifically Γ(n)=(n-1)! for every natural number n.
So with this definition we see that Γ(1/2) = sqrt(pi), so we could say that (-1/2)! = sqrt(pi). It would not actually apply to arranging a number of objects, because you can’t have -1/2 objects.
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When a definition is generalised, the initial meaning of the definition is not necessarily preserved.
Here, we initially defined the factorial to be the number of ways to arrange n objects where n is a **positive integer**. This makes perfect sense. But it only makes sense if n is a positive integer, because we can’t have -1/2 objects.
It turns out that this function has nice properties, like for example the fact that (n+1)! = (n+1)*(n!). It turns out that you can actually define the factorial (on the positive integers) to be the only map f such that f(1) = 1 and f(n+1) = (n+1)*f(n).
This is an equivalent definition of the same function. This alternative definition doesn’t use the fact that we are arranging objects at all. It uses a completely different description of the function. This might allow us to extend this alternative definition to things that are not integers.
It turns out that we can extend the definition to produce a function defined on all real numbers such that f(1) = 1 and f(x+1) = (x+1)*f(x). (you need to add another property to make this extension unique, but that’s another problem)
This new function can be applied to 1/2. However, this function is not defined as the number of ways to arrange x objects. So it’s completely meaningless to interpret it like this.
You are correct, there is no way to express (1/2)! as a number of ways to arrange 1/2 things.
For things other than non-negative integers, the definition of factorial has been extended to something most complicated and less intuitive. n! happens to correspond with permutations when n is an integer >= 0, but it means something else in other cases.
This sort of thing occurs with many mathematical functions. A basic understanding views subtraction as removal. You have 3 cookies, you eat 2, you have 1 cookie left. What happens when you have 3 cookies and eat 4 is meaningless within that framework, you have to extend the definition of subtraction so it means more than just removal in order to talk about negative numbers.
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