The correct answer is option (B):
For John's as well as for Rico's sake
The correct answer is "For John's as well as for Rico's sake." The error lies in the possessive form and the construction "as well as."
The phrase "for John as well as for Rico sake" is grammatically incorrect. When indicating something is done "for the sake of" two individuals, we need to use the possessive form for each name to show that each person individually benefits from the action. Thus, "John's sake" and "Rico's sake" are required.
The structure "as well as" correctly connects the two people. Thus, the correct phrase is "For John's as well as for Rico's sake," indicating that the action was performed for the benefit of both John and Rico.