Probability distributions on a linear vector space over a Galois field and on sets of permutations

We give the exact and limit distributions of the number of vectors from a union of subspaces of the $n$-dimensional vector space $V_n$ over the Galois field $GF(q)$ which enter into a random set of $d$, $1\le d\le n$, linearly independent vectors of this space. We prove that the random variable equal to the number of positions of a random equiprobable permutation which are non-discordant to a $d$-restriction of $m$ pairwise discordant permutations of degree $n$ has in limit, as $n\to\infty$ and $m$ is fixed, the Poisson distribution with parameter $m$. As a consequence we obtain a simple proof of the asymptotic formula for the number of $m\times n$ Latin rectangles where $m$ is fixed and $n\to\infty$.