Question

If 4 married couples are arranged in a row, find the probability that no husband sits next to his wife

Answer 1

Let Ei = the ith couple sitting together where i = 1,2, 3,4

Then :

P(E₁) = (2 × 7!) /8! = (2 × 7!) / (8 × 7!) = 1/4

P(E₁ ∩ E₂) = (2² × 6!) / 8! = 1/14

P(E₁ ∩ E₂ ∩ E₃) = (2³ × 5!) / 8! = 1/42

P(E₁ ∩ E₂ ∩ E₃ ∩ E₄)  = (2⁴ × 4!) / 8! = 1/105

By the inclusion formulae :

P(∩₁⁴E⁴₁) = (⁴₁) P(E₁) - (⁴₂) P(E₁ ∩ E₂) + (⁴₃) P(E₁ ∩ E₂ ∩ E₃) - (⁴₄) P(E₁ ∩ E₂ ∩  E₃ ∩ E₄)

= 4 × 1/4 - 6 × 1/14 + 4 × 1/42 - 1/105 = 1 - 6/14 + 4/42 - 1/105 = 1 - 12/35

1 - 12/35 = 23/35

= Answer 23/35