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 ________ . [2023]

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Solution

Q.
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 ________ . [2023]

Ans.
(43)

$Elements of the type 3 k = 3$

$Elements of type 3 k + 1 = 1, 7, 10$

$Elements of type 3 k + 2 = 2, 5, 11$

$Subsets containing one element S_{1} = 1$

$Subsets containing two elements S_{2} = {}^{3}{C_{1}} \times {}^{3}{C_{1}} = 9$

$Subsets containing three elements S_{3} = {}^{3}{C_{1}} \times {}^{3}{C_{1}} + 1 + 1 = 11$

$Subsets containing four elements S_{4} = {}^{3}{C_{3}} + {}^{3}{C_{3}} + {}^{3}{C_{2}} \times {}^{3}{C_{2}} = 11$

$Subsets containing five elements S_{5} = {}^{3}{C_{2}} + {}^{3}{C_{2}} + {}^{3}{C_{2}} \times 1 = 9$

$Subsets containing six elements S_{6} = 1$

$Subsets containing seven elements S_{7} = 1$

$Required sum = 1 + 9 + 11 + 11 + 9 + 1 + 1 = 43$

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