What is the value of $0 \choose 0$?

Question

It can be 0 according to $0 \choose n$ and 1 according to $n \choose 0$.

Answer 1

Plugging straight into the factorial expression, you wind up with $1$. Combinatorically, I interpret this as "there is always one way to do nothing". Set-theoretically, you can interpret it as there being one function with codomain equal to the empty set, regardless of what the domain is.

Answer 2

Since $\binom n k$ is the number of $k$-element subsets of an $n$-element set, you're asking about the number of empty subsets of the empty set. There's exactly one, so $\binom00=1$.