(In complex analysis it is possible to extend the Gamma function everywhere on the complex plane except the non-positive integers).
So why 0!=1?
this is an assumption. But now the question arise why this assumption was taken?
Its answer is very basic.
1) We know that number of ways of arranging r different things out n different things is =nPr=n!/(n-r)!
2)From fundamental principal of counting we know that number of ways of arranging n different things is n!
But number of ways of arranging n different things must also be equal to nPn (replacing r by n as all things are included)
therefor nPn=n! Or
1/0! = 1
which is only possible if it is asrumed that 0!=1. Hence the assumption was taken.