Why $0!$ is equal to $1$? Many counting formulas involving factorials can make sense for the case $n= 0$ if we define $0!=1 $; e.g., Catalan number and the number of trees with a given number of vetrices. Now here is my question:

If $A$ is an associative and commutative ring, then we can define an
  unary operation on the set of all the finite subsets of our ring,
  denoted by $+ \left(A\right)  $ and $\times \left(A\right)$. While it
  is intuitive to define $+ \left( \emptyset \right) =0$, why should
  the product of zero number of elements be $1$? Does the fact that $0! =1$ have anything to do with 1 being the multiplication unity of integers?


As pointed out in one of the answers to this math.SX question, you can get the Gamma function as an extension of factorials, and then this falls out from it (though this isn't a very combinatorial answer).
