Explanation

(c)
We know, $\lbrack\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}}\mathbf{\ \ }\overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}}\mathbf{\ \ }\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}}\rbrack$
$= \left( \mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}} \right)\mathbf{\bullet}\left\lbrack \left( \overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}} \right)\mathbf{\bullet}\left( \overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}} \right) \right\rbrack$
$= \left( \mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}} \right)\mathbf{\bullet}\left\lbrack \mathbf{\ }\left( \left( \overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}} \right)\mathbf{\bullet}\overrightarrow{\mathbf{b}} \right)\overrightarrow{\mathbf{a}}\mathbf{-}\left( \left( \overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}} \right)\mathbf{\bullet}\overrightarrow{\mathbf{a}}\mathbf{\ } \right)\overrightarrow{\mathbf{b}}\mathbf{\ } \right\rbrack$
$= \left( \mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}} \right)\mathbf{\bullet}\left( \mathbf{\ }\left\lbrack \mathbf{\ }\overrightarrow{\mathbf{c}}\mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ } \right\rbrack\overrightarrow{\mathbf{a}}\mathbf{-}\left\lbrack \mathbf{\ }\overrightarrow{\mathbf{c}}\mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\ } \right\rbrack\mathbf{\ }\overrightarrow{\mathbf{b}} \right)$
$= \left( \mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}} \right)\mathbf{\bullet}\overrightarrow{\mathbf{a}}\left\lbrack \mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\overrightarrow{\mathbf{c}}\mathbf{\ } \right\rbrack - 0$
$= \lbrack\ \overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\mathbf{\ \rbrack\lbrack\ }\overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\ \rbrack$
$= \left\lbrack \ \overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\ \right\rbrack^{2}$