How many 2-dimensional subspaces is a 1-dimensional subspace contained in? V is a  3-dimensional vector space over some field K of order 2.
There are seven 2-dimensional subspaces, and seven 1-dimensional subspaces, using
${n\choose k}_q = \frac{(q^n-1)(q^n-q)...(q^n-q^{k-1})}{(q^k-1)(q^k-q)...(q^k-q^{k-1})}$.
I can show that there each 2-dimensional subspaces contains three 1-dimensional subspaces (using the equation above). 
How do I show that each 1-dimensional subspace is contained within three 2-dimensional subspaces?
Thanks in advance.

Hint 1: You know there are $7 \times 3$ pairs $(X,Y)$ of two dimensional subspaces $X$ including one dimensional subspaces $Y$. You know how many one dimensional spaces there are. Count those pairs another way.
Possible hint 2: There's a duality that switches one and two dimensional subspaces. You might be able to use that to prove what you need.
PS You could change "some field of order 2" to "the field of order 2".
