JEE MAIN - Mathematics (2021 - 27th August Evening Shift - No. 16)
Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A $$ \subseteq $$ S : A $$\ne$$ $$\phi$$ and the sum of all the elements of A is not a multiple of 3} is _______________.
Answer
80
Explanation
3n type $$\to$$ 3, 6, 9 = P
3n $$-$$ 1 type $$\to$$ 2, 5 = Q
3n $$-$$ 2 type $$\to$$ 1, 4 = R
number of subset of S containing one element which are not divisible by 3 = $${}^2$$C1 + $${}^2$$C1 = 4
number of subset of S containing two numbers whose some is not divisible by 3
= $${}^3$$C1 $$\times$$ $${}^2$$C1 + $${}^3$$C1 $$\times$$ $${}^2$$C1 + $${}^2$$C2 + $${}^2$$C2 = 14
number of subsets containing 3 elements whose sum is not divisible by 3
= $${}^3$$C2 $$\times$$ $${}^4$$C1 + ($${}^2$$C2 $$\times$$ $${}^2$$C1)2 + $${}^3$$C1($${}^2$$C2 + $${}^2$$C2) = 22
number of subsets containing 4 elements whose sum is not divisible by 3
= $${}^3$$C3 $$\times$$ $${}^4$$C1 + $${}^3$$C2($${}^2$$C2 + $${}^2$$C2) + ($${}^3$$C1$${}^2$$C1 $$\times$$ $${}^2$$C2)2
= 4 + 6 + 12 = 22
number of subsets of S containing 5 elements whose sum is not divisible by 3.
= $${}^3$$C3($${}^2$$C2 + $${}^2$$C2) + ($${}^3$$C2$${}^2$$C1 $$\times$$ $${}^2$$C2) $$\times$$ 2 = 2 + 12 = 14
number of subsets of S containing 6 elements whose sum is not divisible by 3 = 4
$$\Rightarrow$$ Total subsets of Set A whose sum of digits is not divisible by 3 = 4 + 14 + 22 + 22 + 14 + 4 = 80.
3n $$-$$ 1 type $$\to$$ 2, 5 = Q
3n $$-$$ 2 type $$\to$$ 1, 4 = R
number of subset of S containing one element which are not divisible by 3 = $${}^2$$C1 + $${}^2$$C1 = 4
number of subset of S containing two numbers whose some is not divisible by 3
= $${}^3$$C1 $$\times$$ $${}^2$$C1 + $${}^3$$C1 $$\times$$ $${}^2$$C1 + $${}^2$$C2 + $${}^2$$C2 = 14
number of subsets containing 3 elements whose sum is not divisible by 3
= $${}^3$$C2 $$\times$$ $${}^4$$C1 + ($${}^2$$C2 $$\times$$ $${}^2$$C1)2 + $${}^3$$C1($${}^2$$C2 + $${}^2$$C2) = 22
number of subsets containing 4 elements whose sum is not divisible by 3
= $${}^3$$C3 $$\times$$ $${}^4$$C1 + $${}^3$$C2($${}^2$$C2 + $${}^2$$C2) + ($${}^3$$C1$${}^2$$C1 $$\times$$ $${}^2$$C2)2
= 4 + 6 + 12 = 22
number of subsets of S containing 5 elements whose sum is not divisible by 3.
= $${}^3$$C3($${}^2$$C2 + $${}^2$$C2) + ($${}^3$$C2$${}^2$$C1 $$\times$$ $${}^2$$C2) $$\times$$ 2 = 2 + 12 = 14
number of subsets of S containing 6 elements whose sum is not divisible by 3 = 4
$$\Rightarrow$$ Total subsets of Set A whose sum of digits is not divisible by 3 = 4 + 14 + 22 + 22 + 14 + 4 = 80.
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