  Let $f(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex
planar graph. The order of magnitude of $f(n,P_k)$, where $P_k$ is a path on
$k$ vertices, is $n^{{\lfloor{\frac{k-1}{2}}\rfloor}+1}$. In this paper we
determine the asymptotic value of $f(n,P_5)$ and give conjectures for longer
paths.
