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 _____.

We need to find the number of non-empty subsets of $$S = \{1, 2, 3, 5, 7, 10, 11\}$$ whose element sum is a multiple of 3.

• Residue 0: $$\{3\}$$ — 1 element
• Residue 1: $$\{1, 7, 10\}$$ — 3 elements
• Residue 2: $$\{2, 5, 11\}$$ — 3 elements

Residue 0 group $$\{3\}$$: Including or excluding 3 does not affect the sum mod 3. So this element provides a factor of 2 (include or exclude).

Residue 1 group $$\{1, 7, 10\}$$: Each subset of this group has a sum with some residue mod 3. There are $$2^3 = 8$$ subsets (including empty).

• Sum $$\equiv 0$$: $$\emptyset$$ (sum 0), $$\{1, 7, 10\}$$ (sum 18) — 2 subsets
• Sum $$\equiv 1$$: $$\{1\}$$ (1), $$\{7\}$$ (7), $$\{10\}$$ (10) — 3 subsets
• Sum $$\equiv 2$$: $$\{1,7\}$$ (8), $$\{1,10\}$$ (11), $$\{7,10\}$$ (17) — 3 subsets

Residue 2 group $$\{2, 5, 11\}$$: Similarly, $$2^3 = 8$$ subsets.

• Sum $$\equiv 0$$: $$\emptyset$$ (sum 0), $$\{2, 5, 11\}$$ (sum 18) — 2 subsets
• Sum $$\equiv 1$$: $$\{2, 5\}$$ (7), $$\{2, 11\}$$ (13), $$\{5, 11\}$$ (16) — 3 subsets
• Sum $$\equiv 2$$: $$\{2\}$$ (2), $$\{5\}$$ (5), $$\{11\}$$ (11) — 3 subsets

For the total sum to be $$\equiv 0 \pmod{3}$$, the contributions from residue-1 and residue-2 groups must satisfy:

Valid combinations $$(r_1, r_2)$$:

• $$(0, 0)$$: $$2 \times 2 = 4$$ subsets
• $$(1, 2)$$: $$3 \times 3 = 9$$ subsets
• $$(2, 1)$$: $$3 \times 3 = 9$$ subsets

Total from residue 1 and 2 groups = $$4 + 9 + 9 = 22$$ subsets.

Element 3 can be included or excluded without affecting divisibility by 3:

The answer is $$43$$.