When \epsilon_{ijk} and a_{n} change places the \epsilon_{mni} changes to a cyclic permutation that is still positive and \epsilon_{mni} =\epsilon_{imn}=\epsilon_{nim} but each one of these will give a different final answer.
I don't see how \epsilon_{mni} turns to \epsilon_{imn} when...
Can somebody show me how
\epsilon_{mni}a_{n}(\epsilon_{ijk}b_j c_{k})
Turns in to
\epsilon_{imn}\epsilon_{ijk}a_{n}b_j c_{k}
Something about the first \epsilon I'm not seeing here when the terms are moved around.