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 _____ .

Step-by-step solution

Given set is S = 1 , 2 , 3 , 5 , 7 , 10 , 11 Elements of the type 3 k is 3 . Elements of the type 3 k + 1 are 1 , 7 , 9 . Elements of the type 3 k + 2 are 2 , 5 , 11 Subset containing one element S 1 = 1 Subsets containing two elements S 2 = C 1 3 × C 1 3 = 9 Subsets containing three elements S 3 = C 1 3 × C 1 3 + 1 + 1 = 11 Subsets containing four elements S 4 = C 3 3 + C 3 3 + C 2 3 × C 2 3 = 11 Subsets containing five elements S 5 = C 2 3 × C 2 3 × 1 = 9 Subsets containing six elements S