Coefficient Unravelled

The trinomial expansion is

$\large{ \displaystyle \sum_{k = 0}^{6} \sum_{j = 0}^{6 - k} \left( \begin{array}{c}6 \\ k \end{array} \right) \left( \begin{array}{c}6-k \\ j \end{array} \right) a^{k +6 - j} b^{k + 2 j} c^{12 - 2 k - j} }$

For the term $a^5 b^8 c^5$ , we have $k = 2 , j = 3$ , hence the coefficient is $\displaystyle \left( \begin{array}{c}6 \\ 2 \end{array} \right) \left( \begin{array}{c}4 \\ 3 \end{array} \right) = (15)(4) = \boxed{60}$