The maximum number of 10- and 12-cycles in a planar graph

For a fixed planar graph H, let NP(n,H) denote the maximum number of copies of H in an n-vertex planar graph. In the case when H is a cycle, the asymptotic value of NP(n,Cm) is currently known for m∈{3,4,5,6,8}. In this note, we extend this list by establishing NP(n,C10)∼(n/5)5 and NP(n,C12)∼(n/6)6. We prove this by answering the following question for m∈{5,6}, which is interesting in its own right: which probability mass μ on the edges of some clique maximizes the probability that m independent samples from μ form an m-cycle?