Question

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 _____

Answer 1

Answer: 43

Elements of the type $3k=3$
Elements of the type $3k+1=1,7,9$
Elements of the type $3k+2=2,5,11$
Subsets containing one element $S_1=1$
Subsets containing two elements
$S_2={}^3{C}_1\times {}^3{C}_1=9$
Subsets containing three elements
$S_3={}^3{C}_1\times {}^3{C}_1+1+1=11$
Subsets containing four elements
$S_4={}^3{C}_3+{}^3{C}_3+{}^3{C}_2\times {}^3{C}_2=11$
Subsets containing five elements
$S_5={}^3{C}_2\times {}^3{C}_2\times 1=9$
Subsets containing six elements $S_6=1$
Subsets containing seven elements $S_7=1$
$\Rightarrow \text{ sum }=43$