Question

How many k-dimensional subspaces there are in n-dimensional vector space over fp ?

Answer 1

Answer:

This is given by the Gaussian binomial coefficient:

$\displaystyle\binom{n}{k}_q=\frac{(1-q^n)(1-q^{n-1})\dots(1-q^{n-k+1})}{(1-q)(1-q^2)\dots(1-q^k)}$

where q is the order of the finite field.  So for the field F_p = GF(p) = Z/pZ, just put q=p.