# Counting simultaneous systems of representatives

Many combinatorial problems can be formulated as special cases of the following question: Let $$1 \leq k , and let $$U_1, … ,U_m$$ be $$k$$-dimensional subspaces of $$V := \mathbb{F}_q^d$$, where $$q$$ is a prime power and $$\mathbb{F}_q$$ is a field with $$q$$ elements.

A system of representatives for $$V/U_i$$ is a subset $$Q \subset V$$ with a single element in each equivalence class $$v+U_i$$. The question is: How many such sets $$Q$$ are there that are simultaneously systems of representatives for all of the spaces $$V/U_i$$, where $$1 \leq i \leq m$$?

Is there any theory for this general problem? Is it a known or studied problem in Algebra or Algebraic combinatorics?