Counting Spaces of Intersecting I can't wrap my head around this one.
Let $W$ be an $N$-dimensional vector space over $\mathbb{F}_q$ and $V$ an $l$-dimensional subspace of $W$. Now I want to count the number of subspaces U of dimension $l$ such that $\dim (U\cap V)=l-i$ for a given $i$.
It is clear that I can choose the $(l-i)$-subspace of intersection in $\text{gbc}(l,l-i)=\text{gbc}(l,i)$ ways, where $\text{gbc}$ represents the Gaussian Binomial Coefficient.
After having chosen a $(l-i)$-subspace of intersection I have to complete this space so I will have dimension $l$. I can do this by first adding one of the $q^N-q^l$ vectors in $W$ not already in my $(l-i)$-subspace or $V$. I have however to take into account that some vectors span the same $(l-i+1)$-subspace. Apparently there are $q^l-q^{l-i}$ such vectors. Continuing this process there is 
$$\frac{(q^N-q^l)(q^N-q^{l+1})\dots (q^N-q^{l+i-1})}{(q^l-q^{l-i})(q^l-q^{l-i+1})\cdots (q^l - q^{l-1})}$$
ways of completing my $(l-i)$ subspace.
I don't know why there is $q^l -q^{l-i}$ vectors that span the same subspace. Can anyone explain this? I know that $q^l-q^i$ is the number of vectors missing from my subspace but I'm not really sure what to use this for.
 A: For fixed $U$ and $V$, the denominator counts the number of sequences $(v_1,\ldots,v_i)$ which, taken (as a set) together with a basis for $U\cap V$, form a basis for $V$.  There are $q^l-q^{l-i}$ choices for $v_1$, $q^l-q^{l-i+1}$ choices for $v_2$, etc.
I think a natural way to think about this problem is to start with the analogous problem for subsets. Suppose you have an $N$-element set $W$, and an $l$-element set $V$ contained in it.  You want to count the number of $l$-element sets $U \subset W$ such that $|U\cap V|=l-i$.  The set $U$ decomposes as $l-i$ elements from $V$ together with $i$ elements from $W\setminus V$, and these two collections can be chosen independently.
So the answer is
$$\binom{l}{l-i}\binom{N-l}{i} = \binom{l}{l-i}\binom{N-l}{i}.$$
For your problem we have a similar setup.  We have a decomposition $U\approx (U\cap V) \oplus U/(U\cap V)$.  There are $\binom{l}{l-i}_q$ ways to select $U\cap V$ as a subspace of $V$.  The second term $U/(U\cap V)=U/V$ is [isomorphic to] an $i$-dimensional subspace of $W/V$, and there are $\binom{N-l}{i}_q$ ways to choose this subspace.  So the final count is
$$\binom{l}{l-i}_q\binom{N-l}{i}_q = \binom{l}{l-i}_q\binom{N-l}{i}_q.$$
