Explanation

(c)
Given vectors are coplanar, if $\left| \begin{matrix}
\alpha & 1 & 1 \\
1 & \beta & 1 \\
0 & c & \gamma
\end{matrix} \right| = 0$
Applying $C_{2} \rightarrow C_{2} - C_{1},C_{3} \rightarrow C_{3} - C_{2}$
$\Longrightarrow \left| \begin{matrix}
\alpha & 1 - \alpha & 0 \\
1 & \beta - 1 & 1 - \beta \\
1 & 0 & \gamma - 1
\end{matrix} \right| = 0$
$\Longrightarrow (1 - \alpha)(1 - \beta)(1 - \gamma)\left| \begin{matrix}
\frac{\alpha}{1 - \alpha} & 1 & 0 \\
\frac{1}{1 - \beta} & - 1 & 1 \\
\frac{1}{1 - \gamma} & 0 & - 1
\end{matrix} \right| = 0$
$\Longrightarrow (1 - \alpha)(1 - \beta)(1 - \gamma)\left\lbrack \frac{\alpha}{1 - \alpha}(1) - 1\left( - \frac{1}{1 - \beta} - \frac{1}{1 - \gamma} \right) \right\rbrack = 0$
But α ≠1, β≠1 and γ≠1
$\therefore\frac{1}{(1 - \alpha)} - 1 + \frac{1}{1 - \beta} + \frac{1}{1 - \gamma} = 0$
$\Longrightarrow \frac{1}{1 - \alpha} + \frac{1}{1 - \beta} + \frac{1}{1 - \gamma} = 1$