$\begin{array}{l}
C = \frac{1}{2}\sqrt {216} - 12\sqrt {\frac{3}{2}} + 15\sqrt {\frac{2}{3}} - \sqrt {25 - 4\sqrt 6 } \\
= \frac{1}{2}\sqrt {{6^3}} - 12.\frac{{\sqrt {3.2} }}{2} + 15.\frac{{\sqrt {2.3} }}{3} - \sqrt {{{\left( {2\sqrt 6 } \right)}^2} - 2.2\sqrt 6 + 1} \\
= \frac{{6\sqrt 6 }}{2} - 6\sqrt 6 + 5\sqrt 6 - \sqrt {{{\left( {2\sqrt 6 - 1} \right)}^2}} \\
= 3\sqrt 6 - \sqrt 6 - \left| {2\sqrt 6 - 1} \right|\\
= 2\sqrt 6 - \left( {2\sqrt 6 - 1} \right)\,\,\,\,\left( {do\,\,2\sqrt 6 - 1 > 0} \right)\\
= 2\sqrt 6 - 2\sqrt 6 + 1\\
= 1.
\end{array}$
