# I have been reading Mathematical methods for physicists (sixth edition) by Arfken and Weber and got stuck in section 2.9 Pseudo Tensors,Dual Tensors

In Mathematical methods for physicists (sixth edition) by Arfken and Weber the triple scalar product was defined as (in page 147-148):
For $$\vec A, \vec B, \vec C$$ with components $$A^i, B^j, C^k$$ and a tensor of rank 3 is defined as the direct product of $$\vec A, \vec B, \vec C$$ or $$V^{ijk}=A^iB^jC^k$$. The pseudo-scalar $$V$$ which is the triple product would be,
$$V=\frac{1}{3!}\varepsilon_{ijk}V^{ijk}$$ Where $$\varepsilon_{ijk}$$ is the Levi-civita symbol.
I have calculated for $$V$$ in detail but the factor $$\frac{1}{3!}$$ remains which should not.
Also for 4 dimensional space 4 dimensional Levi-civita $$\varepsilon_{ijkl}$$ was defined in the same page of the book and using this the 4 dimensional volume element is pseudo-scalar is proven as follows:
Let $$\vec A = (dx^0, 0, 0, 0)$$ $$\vec B = (0, dx^1, 0, 0)$$ $$\vec C = (0, 0, dx^2, 0)$$ $$\vec D = (0, 0, 0, dx^3)$$ Now a 4th rank tensor is defined as: $$H^{ijkl}=A^iB^jC^kD^l$$ and the dual pseudo-scalar as: $$H=\frac{1}{4!}\varepsilon_{ijkl}H^{ijkl}$$ And the psedo-scalar is the volume element or $$H=dx^0dx^1dx^2dx^3$$ as written in the book.
I have calculated $$H$$ in detail but the factor $$\frac{1}{4!}$$ remains, which seems to be ignored for some reason.

I also checked that instead of $$V^{ijk}=A^iB^jC^k$$ if we make a anti-symmetric tensor by adding all even permutation of $$ijk$$ and subtracting all odd permutation of $$ijk$$ and set $$V$$ equals to that in that case it gives the correct results. Also in the case of 4-dimensional one.
My question is are factors ignored in the book just to make it to the point or the direct product of vectors is taken to be anti-symmetric by definition.
Please if anyone would like to share some insight that would be great.