# 9. Problem 9.10

In a convex 10-gon \(A_{1} A_{2} \ldots A_{10}\), all sides and all diagonals connecting vertices every other one (i.e., \(A_{1} A_{3}, A_{2} A_{4}\), etc.) are drawn, except for the side \(A_{1} A_{10}\) and the diagonals \(A_{1} A_{9}, A_{2} A_{10}\).

We call a path leading from \(A_{1}\) to \(A_{10}\) a non-self-intersecting broken line (i.e., one in which non-adjacent segments do not have common points) with endpoints \(A_{1}\) and \(A_{10}\), each segment of which coincides with one of the drawn sides or diagonals. Find the number of such paths.
# 9. Problem 9.10

In a convex 10-gon $A_{1} A_{2} \ldots A_{10}$, all sides and all diagonals connecting vertices every other one (i.e., $A_{1} A_{3}, A_{2} A_{4}$, etc.) are drawn, except for the side $A_{1} A_{10}$ and the diagonals $A_{1} A_{9}, A_{2} A_{10}$.

A path from $A_{1}$ to $A_{10}$ is defined as a non-self-intersecting broken line (i.e., one where non-adjacent segments do not have common points) with endpoints $A_{1}$ and $A_{10}$, each segment of which coincides with one of the drawn sides or diagonals. Find the number of such paths.

## Answer: 55