Let $v_1,\cdots,v_n$ be a set of vectors such that the generated vector space has dimension $r$. How many different bases with dimension $r$ can be formed from these vectors? Don't understand the downvote. This is an interesting question. Lets say I am interested out of curiosity.
Edit: To make it more concrete, here is the context, in which I would like to address this question:
Let $p_1,\cdots,p_n$ be the first $n$ primes. Then $\log(p_1),\cdots,\log(p_n)$ are linearly independent over $\mathbb{Q}$. Consider the numbers $1 \cdots n$ and compute the vectors $v_1,\cdots,v_n$ with respect to the basis $\log(p_1),\cdots,\log(p_k)$ where $k = \Pi(n)$ and $\Pi$ is the prime counting function. I would like to know, how many different bases one can form from these vectors. Let $b(n)$ be this number, then for example: $b(2) = 1, b(3)=1 , b(4) = 2, b(5) = 2, b(6) = b(7) = 5$.
Here are the numbers, which form the $5=b(7)$ different bases for $1,\cdots,7$:
[{2, 3, 5, 7}, {2, 5, 6, 7}, {3, 4, 5, 7}, {3, 5, 6, 7}, {4, 5, 6, 7}]
Thanks for your help.