Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 12 / v / Solution

If all $n$ indices are distinct and each lies in $\{1,\dots ,n\}$, their list is a permutation of $(1,\dots ,n)$. Sorting it takes some number $N$ of transpositions. The n-dimensional Levi-Civita symbol equals $(-1{)}^N$ because its sorted component is $1$. ThereforeThe parity of a permutation is independent of the chosen sequence of transpositions.