Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A ⊆ S : A ≠ φ and the sum of all the elements of A is not a multiple of 3} is ______.
3n type → 3, 6, 9 = P
3n – 1 type → 2, 5 = Q
3n – 2 type → 1, 4 = R
number of subset of S containing one element which are not divisible by 3
= 2C1 + 2C1 = 4
number of subset of S containing two numbers whose some is not divisible by 3
= 3C1 X 2C1 + 3C1 x 2C1 + 2C2 + 2C2 = 14
number of subsets containing 3 elements whose sum is not divisible by 3
= 3C2 x 4C1 + (2C2 + 2C1)2 + 3C1(2C2 + 2C2) = 22
number of subsets containing 4 elements whose sum is not divisible by 3
= 3C3 x 4C1 + 3C2(2C2 + 2C2) + (3C12C1 x 2C2)2
= 4 + 6 + 12 = 22.
number of subsets of S containing 5 elements whose sum is not divisible by 3.
= 3C3(2C2 + 2C2) + (3C22C1 x 2C2) x 2
= 2 + 12 = 14
number of subsets of S containing 6 elements whose sum is not divisible by 3 = 4
Total subsets of Set A whose sum of digits is not divisible by 3
= 4 + 14 + 22 + 22 + 14 + 4
= 80