Question #41273

Let $n \ge 2$ be an integer. Consider the determinant $\Delta(y) = \begin{vmatrix} \binom{n}{0}y^2 & \binom{n}{1} & \binom{n}{2} \\ \binom{n+1}{0}y^2 & \binom{n+1}{1} & \binom{n+1}{2} \\ \binom{n+2}{0}y^2 & \binom{n+2}{1} & \binom{n+2}{2} + \frac{1}{y^3} \end{vmatrix}$ Let $T$ be the term independent of $y$ in the binomial expansion of $(\Delta(y))^9$. For an integer $x \in [1, T]$, a matrix $M(x)$ is defined as: $M(x) = \begin{pmatrix} \sin\left(\frac{\pi x}{2}\right) & 1 - \cos(\pi x) & x \sin\left(\frac{\pi x}{2}\right) \\ \sin\left(\frac{\pi x}{3}\right) & \cos\left(\frac{2\pi x}{3}\right) - 1 & \sin\left(\frac{2\pi x}{3}\right) \\ \sin\left(\frac{\pi x}{7}\right) & \sin\left(\frac{2\pi x}{7}\right) & 1 - \cos\left(\frac{2\pi x}{7}\right) \end{pmatrix}$ Find the number of valid values of $x$ such that no row of $M(x)$ is a zero vector.