JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Permutations and Combinations

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: 43

Solution

By remainder on division by

3

, the set splits as

{3}

(residue

0

{1, 7, 10}

(residue

1

) and

{2, 5, 11}

(residue

2

). A subset has sum divisible by

3

exactly when the numbers chosen from the residue-

1

and residue-

2

groups are congruent in count modulo

3

; counting those possibilities and freely including or excluding

3

gives

44

subsets, the empty set among them. Removing the empty set leaves

43

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Integer answer