Step 1: Cross Product:
$\vec b=(1,\lambda,2)$ and $\vec c=(2,3,\lambda)$. Then $\vec b\times\vec c=(\lambda\cdot\lambda-2\cdot3,\ 2\cdot2-1\cdot\lambda,\ 1\cdot3-\lambda\cdot2)=(\lambda^2-6,\ 4-\lambda,\ 3-2\lambda)$.
Step 2: Dot With a:
$\vec a\cdot(\vec b\times\vec c)=\lambda(\lambda^2-6)+(4-\lambda)+(3-2\lambda)=\lambda^3-9\lambda+7$. Coplanarity needs this to be zero.
Step 3: Vieta:
The cubic changes sign three times (values at -4, 0, 2, 3 are -21, 7, -3, 7), so all three roots are real. With no $\lambda^2$ term, their sum is 0. Option (C).
Final Answer:
Option (C).
\[ \boxed{\text{(C) } 0} \]