Answer: B

Video solution:

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Written solution:

For the product to be a multiple of $6$, the two dice together must supply a factor of $2$ and a factor of $3$. The possible sums among the answer choices can occur as follows: $$\begin{gathered} 5=2+3,\quad 7=1+6, \\ 8=2+6,\quad 9=3+6. \end{gathered}$$ There is no way to get sum $6$ while also having a product divisible by $6$: the pairs with sum $6$ are $(1,5),(2,4),(3,3),(4,2),(5,1)$, and none have product divisible by $6$.

Thus, B is the correct answer.