For my own piece of mind(factorials)

Factorial means the amount of ways to arrange n number of objects. So how many ways are there to arrange 0 objects?

Intuition: if you accept that x! = x(x-1)!, then by letting x = 1,
1! = 1(0!)
And then 1! = 1, so
1 = 1(0!)
1 = 0!

In reality, 0! is defined to be 1, and then x! = x(x-1)!, for all other x.

0 because nothing exists there what tf?
you can't arrange what's not there


not understanding any of this ^^

0 because nothing exists there what tf?
you can't arrange what's not there


not understanding any of this ^^

Yes you can. There is one way to arrange them, which is the absence of them.

You can arrange 0 objects in one way.
Follow the pattern
3!=3x2x1
2!=3x2x1/3=2x1
1!=2x1/2=1
0!=1/1=1.
$\displaystyle (n-1)!=\frac{n!}{n}$. If n=1?

So by definition,
x! = 1 if x = 0
x! = x (x-1)! if x > 0
It's defined this way, just like how the trig functions are defined by power series. But if we just take the relation x! = x(x-1)! for intuition:

You know that,
4! = 4 * 3!
3! = 3 * 2!
2! = 2 * 1!
1! = 1 * 0!
and the last line only 'makes sense' if 0! = 1. And like my previous post, we can insert x = 1 into the relation to get
1! = 0!.

https://youtu.be/Mfk_L4Nx2ZI

Watch this for a clearer understanding

If you are not convinced by that then you are not convinced on what factorials really are

Well at least it's starting to sink in