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 "$MOAB$."

[center]Diagram to be added soon[/center]

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.
To determine the maximum number of distinct \( n \)-letter words that can be read from a single \( n \)-gon, we need to consider the different ways an observer can view the vertices of the polygon from outside. The key is to count the number of distinct permutations of the vertices that can be seen from different viewpoints.

1. **Vertices and Edges of the Polygon**:
   - A convex \( n \)-gon has \( n \) vertices and \( n \) edges.

2. **Counting Distinct Permutations**:
   - When an observer stands outside the polygon, they can see the vertices in a specific order. The order depends on the observer's position relative to the polygon.
   - For each pair of vertices, there are two possible ways to see them (one from each side of the line connecting the vertices). This gives \( 2 \binom{n}{2} \) possible pairs.

3. **Using Euler's Formula**:
   - Euler's formula for a planar graph is \( V - E + F = 2 \), where \( V \) is the number of vertices, \( E \) is the number of edges, and \( F \) is the number of faces.
   - For a convex \( n \)-gon, we need to consider the vertices and edges formed by the polygon and the additional vertices and edges formed by the lines connecting the vertices.

4. **Vertices and Edges Outside the Polygon**:
   - Any four vertices of the \( n \)-gon define two vertices outside the polygon. This gives \( 2 \binom{n}{4} \) additional vertices.
   - Together with the \( n \) vertices of the polygon and the point at infinity, this gives:
     \[
     V = 2 \binom{n}{4} + n + 1
     \]
   - Any pair of vertices of the polygon define two edges outside the polygon. Any four vertices of the polygon add four more edges. Together with the \( n \) edges of the polygon, this gives:
     \[
     E = 2 \binom{n}{2} + 4 \binom{n}{4} + n
     \]

5. **Calculating the Number of Faces**:
   - Using Euler's formula \( V - E + F = 2 \), we can solve for \( F \):
     \[
     F = 1 + 2 \binom{n}{2} + 2 \binom{n}{4}
     \]
   - This includes the interior of the polygon, so the number of areas outside the polygon is:
     \[
     2 \binom{n}{2} + 2 \binom{n}{4}
     \]

6. **Conclusion**:
   - The maximum number of distinct \( n \)-letter words that can be read from a single \( n \)-gon is given by the number of distinct permutations of the vertices that can be seen from different viewpoints, which is:
     \[
     2 \binom{n}{2} + 2 \binom{n}{4}
     \]

The final answer is \( \boxed{ 2 \binom{n}{2} + 2 \binom{n}{4} } \)