Question #40599

Let $N$ be the number of ways to select 4 items from a collection of 3 identical apples, 4 identical oranges, and 5 identical bananas. Let $P$ be a plane with normal vector $\vec{n} = \vec{a} \times \vec{b}$, where $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = 2\hat{i} - \hat{j} + \hat{k}$. The plane $P$ satisfies the equation $\vec{r} \cdot \vec{n} = N$. Let $L_1$ be the line $\frac{x-2}{1} = \frac{y+1}{2} = \frac{z-1}{-1}$, and let $L_2$ be the orthogonal projection of the line $L_1$ onto the plane $P$. If $L_3$ is the line $\frac{5x-2}{5} = \frac{5y-4}{5} = \frac{z+1}{1}$, find the shortest distance between the lines $L_2$ and $L_3$.