Number of planar graphs, given an embedding
Pach and Wenger proved in their paper Embedding planar graphs at fixed vertex locations that if $p_1,\ldots,p_n$ are $n$ different points on the plane, then every planar graph on $n$ vertices $v_1,\ldots,v_n$ has a planar embedding in which $v_i$ is found at position $p_i$.