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 \neq φ and B ⋂ C ≠ φ is

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Mathematics

Total Number of Combinations

Question

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 $\neq$ φ and B ⋂ C ≠ φ is

7¹⁰⁰ – 2 · 6¹⁰⁰ + 5¹⁰⁰

6¹⁰⁰ – 2 · 5¹⁰⁰ + 4¹⁰⁰

8¹⁰⁰ – 2 · 7¹⁰⁰ – 6¹⁰⁰

8¹⁰⁰ – 2 · 7¹⁰⁰ + 6¹⁰⁰

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For each elements there are 7 choices without any restriction 7¹⁰⁰.
∴ Number of triplets 7¹⁰⁰ – 6¹⁰⁰ – 6¹⁰⁰ + 5¹⁰⁰= 7¹⁰⁰ – 2 · 6¹⁰⁰ + 5¹⁰⁰.

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