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

Elements of the type $3 \mathrm{k}=3$<br/><br/> Elements of the type $3 \mathrm{k}+1=1,7,9$<br/><br/> Elements of the type $3 \mathrm{k}+2=2,5,11$<br/><br/> Subsets containing one element $S_1=1$<br/><br/> Subsets containing two elements<br/><br/> $S_2={ }^3 C_1 \times{ }^3 C_1=9$<br/><br/> Subsets containing three elements<br/><br/> $\mathrm{S}_3={ }^3 \mathrm{C}_1 \times{ }^3 \mathrm{C}_1+1+1=11$<br/><br/> Subsets containing four elements<br/><br/> $$<br/><br/> \mathrm{S}_4={ }^3 \mathrm{C}_3+{ }^3 \mathrm{C}_3+{ }^3 \mathrm{C}_2 \times{ }^3 \mathrm{C}_2=11 $$<br/><br/> Subsets containing five elements<br/><br/> $\mathrm{S}_5={ }^3 \mathrm{C}_2 \times{ }^3 \mathrm{C}_2 \times 1=9$<br/><br/> Subsets containing six elements $\mathrm{S}_6=1$<br/><br/> Subsets containing seven elements $\mathrm{S}_7=1$<br/><br/> $\Rightarrow \text { sum }=43$