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

The correct answer is 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 S1​=1
Subsets containing two elements
S2​=3C1​×3C1​=9
Subsets containing three elements
S3​=3C1​×3C1​+1+1=11
Subsets containing four elements
S4​=3C3​+3C3​+3C2​×3C2​=11
Subsets containing five elements
S5​=3C2​×3C2​×1=9
Subsets containing six elements S6​=1
Subsets containing seven elements S7​=1
⇒ sum =43