TITLE: The moab Problem
QUESTION [5 upvotes]: The corners of a fixed convex (but not necessarily regular) $n$-gon are labeled with distinct letters. If an observer stands at a point in the plane of the polygon, but outside the polygon, they see the letters in some order from left to right, and they spell a "word" (that is, a string of letters; it doesn't need to be a word in any language). For example, in the diagram below (where $n=4$), an observer at point $X$ would read "$BAMO$," while an observer at point $Y$ would read "$MOAB$."
Diagram to be added soon
Determine, as a formula in terms of $n$, the maximum number of distinct $n$-letter words which may be read in this manner from a single $n$-gon. Do not count words in which some letter is missing because it is directly behind another letter from the viewer's position.
Attemp: I thought about creating the maximum number of regions outside n-gon by extending all the diagonals and sides of n-gon, but it's a bit difficult to get a closed form (I haven't tested it). You have to use V + (F + 1) = E + 2, and that only gets bad from there.

REPLY [3 votes]: Edit: The number of edges in my initial answer was too low. Corrected.
The Euler formula approach works and gives $2{n \choose 2} + 2{n \choose 4} $ areas for a convex $n$-gon $P$ in general position.

Vertices: Any four vertices of $P$ define two vertices outside $P$. Together with the $n$ vertices of $P$ and the point at infinity this gives $V=2{n \choose 4} + n +1$.
Edges: Any pair of vertices of $P$ define two edges outside of $P$. Any four vertices of $P$ add four more edges. Together with the $n$ edges of $P$ this gives $E=2{n \choose 2} + 4{n \choose 4} + n$.
From $V-E+F=2$ it follows that $F=1+ 2{n \choose 2} + 2{n \choose 4} $ including the interior of $P$. So $2{n \choose 2} + 2{n \choose 4} $ areas outside $P$.