If $n\left(A\right)=n$ then $n\left\{(x,y,z):x,y,z\in A,x\ne y,y\ne z,z\ne x\right\}=$

Given that the set A has n elements 

So to fill the ordered pair with the given condition. 

There are $n$ choices for the first coordinate, $\left(n-1\right)$ choices for second coordinate and $\left(n-2\right)$ choices for the third coordinate,

Hence, $n\left\{\{x,y,z\}:x,y,z\in A,x\ne y\ne z\right\}=n(n-1)(n-2)$

Therefore, the number of elements of the given set is $\boxed{n^3-3n^2+2n}$