Holiday Contest 2025 - Individual Round - Problem 9

Ansh has a 10-dimensional hypercube, where every point in the hypercube is a point (x_1,x_2,\dots,x_{10}) with x_i \in \{0,1,2\} for 1 \le i \le 10. He wants to play tic-tac-toe, but he doesn't have any friends. He wants to count the number of ways to get 3 in a row (where space diagonals are allowed), which occurs when the three selected points selects are collinear in 10-dimensional Euclidean space. The number of ways for Ansh to get 3 in a row can be written as 2^a\cdot b, where a and b are positive integers and b is odd. Find the value of a.