Question

Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A ⊆ S : A ≠ `phi` and the sum of all the elements of A is not a multiple of 3} is ______.

विकल्प

• 60

• 70

• 80

• 90

Answer 1

Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A ⊆ S : A ≠ `phi` and the sum of all the elements of A is not a multiple of 3} is 80.

Explanation:

3n type `rightarrow` 3, 6, 9 (divisible by 3)

3n – 1 type `rightarrow` 2, 5 (Not divisible by 3)

3n – 2 type `rightarrow` 1, 4 (Not divisible by 3)

Now, number of subset of S containing one element which are not divisible by 3

= ²C₁ + ²C₁

= 4

Number of subsets of S containing two numbers whose sum is not divisible by 3

= ³C₁ × ²C₁ + ³C₁ × ²C₁ + ²C₂ + ²C₂ 

= 14

Number of subsets containing 3 elements whose sum is not divisible by 3

= ³C₂ × ⁴C₁ + ³C₁ (²C₂ + ²C₂) + ⁴C₃ 

= 22

Number of subsets of S containing 4 elements whose sum is not divisible by 3

= ³C₃ × ⁴C₁ + ³C₂ (²C₂ + ²C₂) + (³C₁ ²C₁ × ²C₂) × 2 

= 4 + 6 + 12

= 22

Number of subsets of S containing 5 elements whose sum is not divisible by 3

= ³C₃ × (²C₂ + ²C₂) + (³C₂ ²C₁ × ²C₂) × 2

= 2 + 12

= 14

Number of subsets of S containing 6 elements whose sum is not divisible by 3

= (³C₃ × ²C₁ × ²C₂) × 2

= 4

⇒ Total subsets of Set A whose sum of digits is not divisible by 3 = 4 + 14 + 22 + 22 + 14 + 4 = 80.