Explanation

(c)
$\left( \overrightarrow{\mathbf{a}} - \overrightarrow{\mathbf{b}} \right) \bullet (\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} - \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}$)
$= \overrightarrow{\mathbf{a}} \bullet \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) - \overrightarrow{\mathbf{b}} \times \left( \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right)$
$= \left\lbrack \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}} \right\rbrack - \left\lbrack \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}}\ \overrightarrow{\mathbf{a}} \right\rbrack = 0$