Let's consider the space $V = k^n$ for $k = \mathbb{F}_p$ where $p$ is a prime number. How many ways are there to choose the basis of this space?
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1This is a standard fact: the answer is $$\prod_{j < n} (p^n-p^j)$$ – Crostul Nov 08 '16 at 10:25
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See also https://math.stackexchange.com/questions/142589/ – Watson Nov 08 '16 at 10:27
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1Note that $$\prod_{j=0}^{n-1}(p^n-p^j)=\prod_{j=0}^{n-1}p^j(p^{n-j}-1)=p^{\binom{n}2}\prod_{j=1}^n(p^j-1);,$$ the last form also sometimes being useful. – Brian M. Scott Nov 08 '16 at 10:39