Statement A: The number of ways in which 3 married couples and their 4 children can be seated in a row such that no husband and wife sit adjacent to each other equals 10! - 3*9! + 3*8! - 7!. Statement R: The number of ways in which at least one of three events A, B, C occurs is n(A union B

Question

Statement A: The number of ways in which 3 married couples and their 4 children can be seated in a row such that no husband and wife sit adjacent to each other equals 10! - 3*9! + 3*8! - 7!. Statement R: The number of ways in which at least one of three events A, B, C occurs is n(A union B union C) = n(A) + n(B) + n(C) - n(A intersection B) - n(B intersection C) - n(C intersection A) + n(A intersection B intersection C).

Accepted Answer

Both A and R are correct and R is the correct explanation of A. Total arrangements = 10!. Let Ei = husband i and wife i are adjacent. Treat them as one block: |Ei| = 2*9!, |Ei intersection Ej| = 4*8!, |E1 intersection E2 intersection E3| = 8*7!. By inclusion-exclusion, arrangements with at least one couple together = 3*(2*9!) - 3*(4*8!) + 8*7! = 6*9! - 12*8! + 8*7!. Desired = 10! - 6*9! + 12*8! - 8*7! = 10! - 3*(2*9!) + 3*(4*8!) - 8*7!. This matches the form 10! - 3*9! only if the '2' factor is absorbed into the 9! term with a different grouping; the standard form is 10! - 3*9! + 3*8! - 7! after dividing out common factors... the exact form depends on treating the pair as an ordered unit. St

Solution

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