I understand that the factorial gives the number of arrangements. For example: the factorial of zero i.e. an empty set ( doesn't occur) is 1.
As the empty set can be arranged only in 1 way – i.e. by filling nothing. Now, let's take an example: 5 distinct seats. How many ways 5 distinct seats can be arranged? – 5! ways i.e. 120.
So, basically, factorial gives us the arrangements. Now, the question is why do we need to know the factorial of a negative number?, let's say -5. How can we imagine that there are -5 seats, and we need to arrange it? Something, which doesn't exist shouldn't have an arrangement right? Can someone please throw some light on it?. I saw some research done on the formula – https://mathoverflow.net/questions/10124/the-factorial-of-1-2-3
Best Answer
The factorials of negative integers have no defined meaning.
Reason: We know that factorials satisfy $x\cdot(x-1)!=x!$. However, if there was a $(-1)!$, then we'd be able to write: \begin{align} x\cdot(x-1)!&=x!\\ 0\cdot(-1)!&=0!\\ 0&=1 \end{align} Contradiction.
However, there is a meaningful definition of the factorials of non-integers! Here is a graph. One motivation for this particular way of doing it is, they wanted these two properties to hold (even when $x,t$ are not integers):
$x\cdot(x-1)!=x!$ for all $x$ (as long as $(x-1)!$ exists)
$(n+t)!\approx n!\,n^t$ (when $n$ is large)