Let be $\mathbb{F}_{q}^{n}$ is field of dimension - $q^n$. Let $C$ be linear code of $dimC = k$. We have three questions :
1) Number of basises in $C$ ($k$ - dimensional linear subspace)?
2) Number of different linear codes in $\mathbb{F}_q^n$?
3) Number of different generator matrices for code of $dim = k$ in $\mathbb{F}_q^n$?
My attempt :
1) Let $C$ be a linear code in $\mathbb{F}_q^n$ with $dim = k$. Then number of basises in $C$ is $(q^k-1)\dots(q^k - q^{k-1})$.
2) Number of linear codes is number of different $k-$dimensional subspaces in $\mathbb{F}_q^n$. It should be $\dfrac{(q^n-1)\dots (q^n-q^{k-1})}{(q^k-1)\dots(q^k - q^{k-1})}$.
3) Number of generator matrices is $(q^n-1) \dots (q^n - q^{k-1})$.
Am I right?
For $1)$, you're counting different orders of choosing the basis as different. Since a basis defined as a set, not a tuple, you need to divide by $k!$.
Your result for $2)$ is correct, since you've divided the number of “basis $k$-tuples” in $\mathbb F_q^n$ by the one within the resulting subspaces, so the factors of $k!$ cancel in this case.
In $3)$, the question is unclear; it could be read to ask for the number of different generator matrices for a particular code of dimension $k$ in $\mathbb F_q^n$, or for any such code. Your result is correct for the latter interpretation.