Explanation:

A. (III) (II) (V) (I) (IV): This arrangement forms the sentence: "Extended families help keep the children in the family healthier, allowing parents to continue and pursue careers." This sequence is grammatically correct and logically coherent.\nB. (I) (IV) (II) (V) (III): This arrangement results in an incoherent sentence: "Allowing parents to continue and pursue careers help keep the children in the family healthier extended families."\nC. (I) (II) (V) (III) (IV): This arrangement also results in an incoherent sentence: "Allowing parents to continue help keep the children in the family healthier extended families and pursue careers."\nD. (III) (I) (IV) (II) (V): This arrangement forms: "Extended families allowing parents to continue and pursue careers help keep the children in the family healthier." While grammatically possible, the structure is less direct and clear than option A, as the subject-verb relationship for 'help' becomes slightly convoluted.