The value of the determinant […] is :

The correct answer is Option (3) → $a+b+c+1$

$Δ=\begin{vmatrix} a+1 & b & c \\a & b+1 & c \\a & b & c+1 \end{vmatrix}$

$C_1→C_1+C_2+C_3$

$Δ=\begin{vmatrix} a+b+c+1 & b & c \\a+b+c+1 & b+1 & c \\a+b+c+1 & b & c+1 \end{vmatrix}$

$Δ=(a+b+c+1)\begin{vmatrix} 1 & b & c \\1 & b+1 & c \\1 & b & c+1 \end{vmatrix}$

$R_2→R_2-R_1$

$R_3→R_3-R_1$

$Δ=(a+b+c+1)\begin{vmatrix} 1 & b & c \\0 & 1 & 0 \\0 & 0 & 1 \end{vmatrix}$

$Δ=a+b+c+1$