<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Given a convex 20-sided polygon \( P \), divide \( P \) into 18 triangles using 17 diagonals that do not intersect inside \( P \). The resulting figure is called a triangulation of \( P \). For any triangulation \( T \) of \( P \), the 20 edges of \( P \) and the 17 additional diagonals are referred to as the edges of \( T \). A set of any 10 edges of \( T \) that have no common endpoints is called a perfect matching of \( T \). Find the maximum number of such perfect matchings \( T \) has over all triangulations of \( P \). Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
55
Given a convex 20-sided polygon \( P \), divide \( P \) into 18 triangles using 17 diagonals that do not intersect inside \( P \). The resulting figure is called a triangulation of \( P \). For any triangulation \( T \) of \( P \), the 20 edges of \( P \) and the 17 additional diagonals are referred to as the edges of \( T \). A set of any 10 edges of \( T \) that have no common endpoints is called a perfect matching of \( T \). Find the maximum number of such perfect matchings \( T \) has over all triangulations of \( P \).
test