$9.17 C_{x}^{x-1}+C_{x}^{x-2}+C_{x}^{x-3}+\ldots+C_{x}^{x-9}+C_{x}^{x-10}=1023$.
9.17 Obviously, when $x=10$ the left side of the equation is 1 less than the sum of the binomial coefficients in the expansion of the binomial $(a+b)^{10}$. This sum is 1024. Therefore, $x=10$ is a solution to the equation. There are no other solutions, as for $x>10$ the left side of the equation is greater than 1023.

Answer: $x=10$.