Question #41812

Let $\vec{u} = \sec^2 \theta \hat{i} + \cot^2 \theta \hat{j} + \csc^2 \theta \hat{k}$, $\vec{v} = \tan^2 \theta \hat{i} + \csc^2 \theta \hat{j} + \sec^2 \theta \hat{k}$, and $\vec{w} = \hat{i} - \hat{j} + (\cot^2 \theta - \tan^2 \theta)\hat{k}$, where $\theta \in \left(0, \frac{\pi}{4}\right)$. Let $p$ be the volume of the parallelepiped whose coterminous edges are $\vec{u}, \vec{v}, \vec{w}$. A biased coin, which shows heads with probability $p$, is tossed 10 times. Let $X$ denote the number of heads obtained. Find the value of $\sum_{k=0}^{10} (k^2 - 3k + 2) P(X=k) + 2024^p$