# Exercise about the number of subspaces of a certain dimension

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.