= Solution
If all $n$ indices are distinct and each lies in $\{1,\ldots,n\}$, their list is a <permutation> of $(1,\ldots,n)$. Sorting it takes some number $N$ of transpositions. The <n-dimensional Levi-Civita symbol> equals $(-1)^N$ because its sorted component is $1$. Therefore
$$
\boxed{\epsilon_{i_1\cdots i_n}=\begin{cases}
0,&\text{some indices repeat},\\
+1,&(i_1,\ldots,i_n)\text{ is an even permutation},\\
-1,&(i_1,\ldots,i_n)\text{ is an odd permutation}.
\end{cases}}
$$
The <parity of a permutation> is independent of the chosen sequence of transpositions.
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