Number of $l$-dimensional subspaces of a vector space containing a given subspace

Question

What is the number of $l$ dimensional subspaces of $V$ containing a given $k$ dimensional subspace?

Where $F$ is a finite field with $q$ elements and $V$ is an $n$ dimensional vector space over $F$.

Answer 1

Let $U$ denote our $k$-dimensional space, and let $W$ denote the $l$-dimensional space which contains it.

Hint: There is a one-to-one correspondence between the spaces $W$ containing $U$ and the quotient spaces $W/U$, each of which is an $l-k$ dimensional subspace of $V/U$.