  Kochen-Specker (KS) theorem denies the possibility for the noncontextual
hidden variable theories to reproduce the predictions of quantum mechanics. A
set of projection operators (projectors) and bases used to show the
impossibility of noncontextual definite values assignment is named as the KS
set. Since one KS set with a mixture of 16 rank-1 projectors and 14 rank-2
projectors proposed in 1995 [Kernaghan M and Peres A 1995 Phys. Lett.\ A
\textbf{198} 1] for a three-qubit system, there are plenty of the same type KS
sets and we propose a systematic way to produce them. We also propose a
probabilistic state-dependent proof of the KS theorem that mainly focuses on
the values assignment for all the rank-2 projectors.
