The number of triplets (A, B, C) where A, B, C are subsets of X= 1,2,3, ldots ldots, 100 such that A ∩ B ∩ C=φ but A ∩ B ≠ φ and B ∩ C ≠ φ is

Question

The number of triplets (A, B, C) where A, B, C are subsets of X= 1,2,3, ldots ldots, 100 such that A ∩ B ∩ C=φ but A ∩ B ≠ φ and B ∩ C ≠ φ is (A) 7100-2 ⋅ 6100+5100 (B) 8100-2 ⋅ 7100+6100 (C) 6100-2 ⋅ 5100+4100 (D) 8100-2 ⋅ 7100-6100

Tardigrade Admin · Accepted Answer

For each elements there are 7 choices without any restriction 7100.

Q. The number of triplets $(A, B, C)$ where $A, B, C$ are subsets of $X = {1, 2, 3, \dots\dots, 100}$ such that $A \cap B \cap C = ϕ$ but $A \cap B \neq = ϕ$ and $B \cap C \neq = ϕ$ is

$7^{100} - 2 \cdot 6^{100} + 5^{100}$

$8^{100} - 2 \cdot 7^{100} + 6^{100}$

$6^{100} - 2 \cdot 5^{100} + 4^{100}$

$8^{100} - 2 \cdot 7^{100} - 6^{100}$

For each elements there are 7 choices without any restriction $7^{100}$ .

Q. The number of triplets (A,B,C) where A,B,C are subsets of X={1,2,3,……,100} such that A∩B∩C=ϕ but A∩B=ϕ and B∩C=ϕ is

7100−2⋅6100+5100

8100−2⋅7100+6100

6100−2⋅5100+4100

8100−2⋅7100−6100