$K=\mathbb{Z}_p$ for some prime p, and $dim V = n$.
It has been shown that the number of different bases in $V$ is:
$\frac{1}{n!} \prod_{i=0}^{n-1}(p^n - p^i)$ (bases which are permutations of one another are regarded as equivalent)
Derive from this that an $n x n$ matrix with elements in $\mathbb{Z}_p$ chosen independently at random has nonzero determinant with probability equal to $\prod_{i=1}^{n}(1-p^{-i})$
Not quite sure where to start.