$$ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^5 x \cos ^7 x d x= $$

A batch of $\mathbf{1 0}$ cupcakes consists of $\mathbf{5}$ chocolate, $\mathbf{3}$ vanilla, and $\mathbf{2}$ strawberry. If 4 cupcakes are selected to be put into a gift box, find the number of different ways they can be chosen if the selection must include at least $\mathbf{2}$ chocolate, at most $\mathbf{1}$ vanilla, and exactly $\mathbf{1}$ strawberry cupcake.

If a straight line passing through a fixed point $(a, b)$, where $\boldsymbol{a}, \boldsymbol{b}>\mathbf{0}$, makes positive intercepts OA and OB on the coordinate axes, then the least value of $\mathbf{O A}+\mathbf{O B}$ is:

Let $\mathbf{P}$ be a point on the line $L_1: \frac{x-2}{2}=y+1=\frac{z-1}{2}$ such that its distance from the point $A(2,-1,1)$ is 6 units.

Given that $\boldsymbol{x}$-coordinate of $\mathbf{P}$ is greater than $\mathbf{2}$,

Find the coordinates of point Q on the line $L_2: x-1=\frac{y-2}{2}=\frac{z-2}{2}$ such that $\mathbf{Q}$ is the closest point to $\mathbf{P}$.