Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 4 / 1E / Solution

$|V|=p^n$. An ordered basis is chosen successively in $(p^n-1)(p^n-p)\cdots (p^n-p^{n-1})$ ways, which also counts bijective endomorphisms. Counting ordered independent $k$-tuples and dividing by the number of bases of ${\mathbb{F}}_p^k$ gives the Gaussian binomial ${\prod }_{i=0}^{k-1}(p^n-p^i)/(p^k-p^i)$.