Let N denote the number that turns up when a fair die is rolled. If...

Given equations are

$x + y + z = 12x + Ny + 2z = 23x + 3y + Nz = 3$

So,

∆ = \left|\begin{cases}1 & 1 & 1 \\ 2 & N & 2 \\ 3 & 3 & N\end{cases}\right|

Applying ${C}_{2} ↔ {C}_{2} - {C}_{1} and${C}_{3} ↔ {C}_{3} - {C}_{1}$</p><p>$⇒ ∆ = \left|$\begin{cases}1 & 0 & 0 \\ 2 & N - 2 & 0 \\ 3 & 0 & N - 3\end{cases}$\right|$</p><p>$⇒ ∆ = \left(N - 2\right)\left(N - 3\right)$</p><p>For unique solution$∆ \ne 0$, so</p><p>$N \ne 2, 3$</p><p>Also,$N \in \left\{1, 2, 3, 4, 5, 6\right\}$and for unique solution$N$can take values \left\{1, 4, 5, 6\right\}$.

$⇒ P$(system has unique solution ) $= \dfrac{4}{6} = \dfrac{k}{6}$

So, $k = 4$

Therefore, required sum is

$= 4 + 1 + 4 + 5 + 6 = 20$