Let $S=\left\{1,2,3,5,7,10,11\right\}$. The number of non-empty subsets of $S$ that have the sum of all elements a multiple of 3, is ……. 

$\mathrm{S}=\{1,2,3,5,7,10,11\}=\left\{3\right\}\cup \{1,7,10\}\cup \{2,5,11\}=\mathrm{A}\cup \mathrm{B}\cup \mathrm{C}$
$\mathrm{Let}\mathrm{X}\subseteq \mathrm{S},\mathrm{X}\ne \mathrm{\Phi }$
$\begin{array}{cccc} & \in \mathrm{A}& \in \mathrm{B}& \in \mathrm{C}\\ \begin{array}{l}\mathrm{Number}\mathrm{of}\\ \mathrm{elements}\mathrm{X}\end{array}& 0\mathrm{or}1& 0& 0\\ & & 1& 1\\ & & 2& 2\\ & & 3& 3\\ & & 3& 0\\ & & 0& 3\end{array}$
$\text{Ans=2×}{\text{[\hspace{0.17em}}}^3{\mathrm{C}}_0{.}^3{\mathrm{C}}_0+{\mathrm{\hspace{0.17em}}}^3{\mathrm{C}}_1{.}^3{\mathrm{C}}_1+{\mathrm{\hspace{0.17em}}}^3{\mathrm{C}}_2{.}^3{\mathrm{C}}_2+{\mathrm{\hspace{0.17em}}}^3{\mathrm{C}}_3{.}^3{\mathrm{C}}_3+{\mathrm{\hspace{0.17em}}}^3{\mathrm{C}}_3{.}^3{\mathrm{C}}_0+{\mathrm{\hspace{0.17em}}}^3{\mathrm{C}}_0{.}^3{\mathrm{C}}_3]-1=43$