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

Step-by-step solution

We can categorise the numbers of set S = 3 , 6 , 9 ⏟   3 n  type ,   2 , 5 ⏟   3 ( n - 1 )  type     , 1 , 4 ⏟   3 ( n - 2 )  type Let N p = Number of subsets of S containing p elements which are not divisible by = 3 For p = 1 n ( N 1 ) = C 1 2 + C 1 2 = 4 For p = 2 n ( N 2 ) = C 1 3 C 1 2 + C 1 3 C 1 2 + C 2 2 + C 2 2 = 14 For p = 3 n ( N 3 ) = C 1 3 C 2 2 + C 2 2 + C 2 3 C 1 2 + C 1 2 + C 2 2 C 1 2 + C 1 2 C 2 2 = 22 For p = 4 n ( N 4 ) = C 1 3 C 2 2