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.