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?