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Let $p$ be a prime number and $V$ an $n$-dimensional space over $\mathbb{F}_p$. For $r \le n,r \in \mathbb{N}$ find the number of $r$-dimensional subspaces of $V$.

I suspect we can choose an $r$-dimensional subspace in $\binom{n}{r}$ ways. Is this reasoning correct?

Zelazny
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    If you're picking your subspaces amongst the subspaces spanned by the standard basis vectors, then yes. But not all subspaces are spanned by these vectors. Which means you first have to find the number of bases of $V\simeq\mathbb F_p^n$ that can be constructed... which would be exactly the number of invertible matrices in $\mathcal M_n(\mathbb F_p)$. – ManifoldFR Nov 17 '16 at 22:32
  • @Groovy. I am able to find the number of bases Ben mentioned. I fail to see one thing. In finite fields, does each subspace have an individual set of spanning vectors? – Zelazny Nov 17 '16 at 22:41

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