Compute the three building blocks.

$2\vec{p}+\vec{q}=(4+1)\hat{i}+(2-1)\hat{j}+(6+1)\hat{k}=5\hat{i}+\hat{j}+7\hat{k}.$

$\vec{p}-2\vec{q}=(2-2)\hat{i}+(1+2)\hat{j}+(3-2)\hat{k}=0\hat{i}+3\hat{j}+\hat{k}.$

$\vec{p}\times \vec{q}=\left|\begin{array}{c}\hat{i}\hat{j}\hat{k}\\ 213\\ 1-11\end{array}\right|=\hat{i}(1+3)-\hat{j}(2-3)+\hat{k}(-2-1)=4\hat{i}+\hat{j}-3\hat{k}.$

Matching components on each side of $15\hat{i}+10\hat{j}+6\hat{k}=\alpha (5,1,7)+\beta (0,3,1)+\gamma (4,1,-3)$:

$5\alpha +4\gamma =15,\alpha +3\beta +\gamma =10,7\alpha +\beta -3\gamma =6.$

Solving this linear system yields $\alpha =\frac{7}{5}$, $\beta =\frac{11}{5}$, $\gamma =2$.

$\therefore \gamma =2.$