= Solution
{c}
Put $L=\operatorname{span}\{v\}$. Every two-dimensional <vector subspace> containing $v$ is $W=\operatorname{span}\{v,w\}$ for some $w\notin L$. There are $p^n-p$ choices for $w$ in $V$. Within any one such $W$, exactly $p^2-p$ choices of $w\in W\setminus L$ yield that same $W$: each forms a <linearly independent> pair with $v$, and that pair is a <basis> of $W$.
These sets of choices partition $V\setminus L$, because $w$ determines $\operatorname{span}\{v,w\}$ uniquely. Thus the number is
$$
\boxed{\frac{p^n-p}{p^2-p}=\frac{p^{n-1}-1}{p-1}.}
$$
Equivalently these <vector subspaces> correspond to the one-dimensional <vector subspaces> of the <quotient vector space> $V/L$. For $n=1$ there are none and the formula gives zero.
Back to article page
