Evaluate $\Delta = \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix}$.

$(a - b)(b - c)(c - a)$

The correct answer is Option (2) → $(a - b)(b - c)(c - a)$ ##

Applying $R_2 \rightarrow R_2 - R_1$ and $R_3 \rightarrow R_3 - R_1$, we get

$\Delta = \begin{vmatrix} 1 & a & bc \\ 0 & b - a & c(a - b) \\ 0 & c - a & b(a - c) \end{vmatrix}$

Taking factors $(b - a)$ and $(c - a)$ common from $R_2$ and $R_3$, respectively, we get

$\Delta = (b - a)(c - a) \begin{vmatrix} 1 & a & bc \\ 0 & 1 & -c \\ 0 & 1 & -b \end{vmatrix}$

$= (b - a)(c - a) [(-b + c)] \text{ (Expanding along first column)}$

$= (a - b)(b - c)(c - a)$