What is $0^0$? Indeterminate or 1?

Question

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Zero to zero power

Sorry for asking this simple question, but googling this question yields conflicting answers.
Some say it's indeterminate, other's say it's $1$.

Answer 1

The short answer: 'yes'. The slightly longer answer: $0^0$ has different meanings in different contexts, depending on whether the power represents the analytical power (defined either through continuation of the function $x^y$ from rational values of $y$ to real values) or the combinatorial power (where $A^B$ is defined as the number of maps from the set $A$ to the set $B$). The 'analytical' $0^0$ is considered undefined because the double-limit $\displaystyle\lim_{x,y\to 0+}x^y$ will yield different values (in fact, it can yield all possible values) depending on what path is taken towards the origin. By contrast, the combinatorial power $0^0$ is taken to be $1$; there is considered to be one map from the empty set to itself, the identity map. This convention substantially simplifies formulas of a combinatorial nature; for instance, the binomial identity $(1+x)^n = x^0+nx^1+{n \choose 2}x^2+\ldots$.

ETA: I called the combinatorial $0^0=1$ this a 'convention' and said 'there is considered to be', but looking back on this I think I wasn't strong enough here; by the definition of a map or function from one set to another (that is, a set $S$ of ordered pairs $\langle a,b\rangle, a\in A, b\in B$ such that for each $a\in A$ there's exactly one pair $\langle\alpha,\beta\rangle\in S$ with $\alpha=a$ and each element of $S$ corresponds to a single element of $A$) this statement holds; in fact, this gives $n^0=1$ for all $n$ including zero, since when $A$ is the empty set the condition is vacuously satisfied iff $S$ is also the empty set.