Let $V$ be a vector space of dimension $n$ over $\mathbb{F}_q$, and let $U$ be a subspace of dimension $k$. I want to compute the number of subspaces $W$ of $V$ of dimension $m$ such that $W\cap U=0$.
I know why the number of subspaces of $V$ that contain $U$ and have dimension $m$ is $\binom{n-k}{m-k}_q$, but I don't understand why $q^{km}\binom{n-k}{m}_q$ is number of these subspaces?