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Question: If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are unit vectors representing evolutionary traits, find the maximum value of $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$ under positive scalar constraints.
Solution: The scalar triple product equals the volume of the parallelepiped. Maximum occurs when vectors are orthogonal, giving $|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| = 1 \cdot 1 \cdot 1 \cdot \sin(90^\circ)\cos(0^\circ) = 1$.
\boxed{1}Question: A research scientist is preparing 7 different chemical solutions. How many ways can they distribute these to 3 identical storage vials, ensuring each vial contains at least one solution?

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