MHB Finite Group Inverses: Proving $N_{ABC}=N_{CBA}$

[Fallen Angel]

Hi,

I bring a new algebraic challenge ;)

Let $G$ be a finite group and $U,V,W\subset G$ arbitrary subsets of $G$.
We will denote $N_{UVW}$ the number of triples $(x,y,z)\in U\times V \times W$ such that $xyz$ is the unity of $G$, say $e$.
Now suppose we have three pairwise disjoint sets $A,B,C$ such that $G=A\cup B \cup C$

Prove that $N_{ABC}=N_{CBA}$.

 

[Fallen Angel]

A hint:

Spoiler

Start proving that for arbitrary $U,V\subset G$
$N_{UVG}=|U||V|$
and for arbitraty $U,V,W\subset G$
$N_{UVW}=N_{WUV}=N_{VWU}$