The number of all possible value(s) of $k$ for which the system of equations $k x + y + z = k - 1, x + k y + z = k - 1, x + y + k z = k - 1$ has no solution is

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Solution

$k x + y + z = k - 1$
$x + k y + z = k - 1$
$x + y + k z = k - 1$
The system has no solution if $Δ = 0$ and one of $Δ_{1}, Δ_{2}, Δ_{3}$ is non-zero.
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} k & 1 & 1 1 & k & 1 1 & 1 & k \end{matrix} ∣ ∣ ∣ ∣ = 0$

$\Rightarrow k (k^{2} - 1) - (k - 1) + (1 - k) = 0$
$\Rightarrow (k - 1) (k (k + 1) - 1 - 1) = 0$
$\Rightarrow (k - 1) (k^{2} + k - 2) = 0$
$\Rightarrow (k - 1)^{2} (k + 2) = 0$
$\Rightarrow k = 1, - 2$
But for $k = 1,$ the system has infinite solutions.
$∴ k = - 2$

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