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Number of subsets of a given set whose sum of elements is (or is not) a multiple of 3

The same question, asked 2 times: 2021 Paper 21*, 25 Jan 2023. Every wording below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 21

    85
    Let S={1,2,3,4,5,6,9}S\mathbf{= \{ 1,2,3,4,5,6,9\}}. Then the number of elements in the set T={A⊆S:A≠ϕ\mathbf{T = \{ A \subseteq S:A \neq}\phi and the sum of all the elements of AA is not a multiple of 3}\} is

  2. Question 2

    25 Jan 2023

    83
    Let S={1,2,3,5,7,10,11}\mathbf{S = \{ 1,2,3,5,7,10,11\}}. The number of non-empty subsets of SS that have the sum of all elements a multiple of 3, is