Exercise about the number of subspaces of a certain dimension

Question

I have the following exercise:

Let $\mathbb{F}$ be a finite field and $q = |\mathbb{F}|$ the number of elements of this field. Determine the number of $d$-dimensional subspaces of $\mathbb{F}^n$. That means, look at $$Gr_d(\mathbb{F}^n) := \{ U \subset \mathbb{F}^n \mid U \text{ subspace with } \dim(U) = d\}$$ and determine $|Gr_d(\mathbb{F}^n)|$. For doing this, inspect the obvious group action of $Gl_n(\mathbb{F})$ on the set $Gr_d(\mathbb{F}^n)$.

I think the group action that is mentioned is the following:

$.:Gl_n(\mathbb{F})\times Gr_d(\mathbb{F}^n) \to Gr_d(\mathbb{F}^n), (F, U) \mapsto F(U)$

So what can I do here? I think I will need the orbit-stabilizer theorem or the Lagrange theorem.

Thank you for your help.

Regards, S. M. Roch

Answer 1

This reference (http://math.columbia.edu/~nsnyder/Solutions1.pdf) provides two proofs, one of the same style as in (How to count number of bases and subspaces of a given dimension in a vector space over a finite field?), the other directly linked to the method that you have to use.