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

Elements of the type $3 k =3$
Elements of the type $3 k +1=1,7,9$
Elements of the type $3 k +2=2,5,11$
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 $S _7=1$
$\Rightarrow \text { sum }=43$