Let S - 1- 2- 3- 4- 5- 6- 9- Then the number of elements in the set T - A - S - A - and the sum of all the elements of A is not a multiple of 3 is

There are two numbers of the type 3λ + 1, two numbers of nbsp;the type 3λ – 1 and three numbers of the type 3λ. So, number of subsets whose sum is divisible b ...

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Standard XII

Mathematics

Subset and Superset

Let S = 1, 2,...

Question

Let $S = {1, 2, 3, 4, 5, 6, 9}$ . Then the number of elements in the set $T = {A \subseteq S : A \neq ϕ$ and the sum of all the elements of $A$ is not a multiple of $3}$ is

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Solution

There are two numbers of the type $3 λ + 1,$ two numbers of the type $3 λ - 1$ and three numbers of the type $3 λ$ .
So, number of subsets whose sum is divisible by 3
$= 2^{3} \cdot (^{2} C_{0}^{2} +^{2} C_{1}^{2} +^{2} C_{2}^{2})$
$= 48$
Required number of subsets $= 2^{7} - 48 = 80$

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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

There are two numbers of the type 3λ+1, two numbers of the type 3λ–1 and three numbers of the type 3λ. So, number of subsets whose sum is divisible by 3 =23⋅(2C20+2C21+2C22) =48 Required number of subsets =27−48=80

Let $S = {1, 2, 3, 4, 5, 6, 9}$ . Then the number of elements in the set $T = {A \subseteq S : A \neq ϕ$ and the sum of all the elements of $A$ is not a multiple of $3}$ is

There are two numbers of the type $3 λ + 1,$ two numbers of the type $3 λ - 1$ and three numbers of the type $3 λ$ .
So, number of subsets whose sum is divisible by 3
$= 2^{3} \cdot (^{2} C_{0}^{2} +^{2} C_{1}^{2} +^{2} C_{2}^{2})$
$= 48$
Required number of subsets $= 2^{7} - 48 = 80$