$A=\begin{bmatrix} 1 &1 &1 \\ 2 & 3 & 4\\ 4 & 5 &6 \end{bmatrix}$
Augmented matrix
$\left [ A \vdots B \right ]=\begin{bmatrix} 1 &1 &1 \ \ \vdots &1\\ 2 & 3 & 4 \ \ \vdots &1\\ 4 & 5 &6 \ \ \vdots &2 \end{bmatrix}$
$\left [ A \vdots B \right ]=\begin{bmatrix} 1 &1 &1 \ \ \vdots &1\\ 2 & 3 & 4 \ \ \vdots &1\\ 2 & 2 &2\ \ \vdots &1 \end{bmatrix} \ \ \ \ \ \ \ \ R_{3} \rightarrow R_{3}-R_{2}$
$\left [ A \vdots B \right ]=\begin{bmatrix} 1 &1 &1 \ \ \vdots &1\\ 2 & 3 & 4 \ \ \vdots &1\\ 0 & 0 &0\ \ \vdots &-1 \end{bmatrix} \ \ \ \ \ \ \ \ R_{3} \rightarrow R_{3}- 2 \times R_{1}$
Here rank of matrix $A$ and rank of augmented matrix $ \left [ A \vdots B \right ] $ is not equal.
So, The system of equation has No Solution.
Hence, Option(C)No Solution.