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It's not that A is grammatically wrong- it's grammatically fine- it's just that it means something different from C. What is the sentence trying to say? That the FBI is investigating anyone with a financial relationship to the Mafia, even if that financial relationship is distant. In A, 'distant' is an adjective- it describes 'anyone'. The sentence in A means that the FBI is investigating anyone with a financial relationship with the Mafia, even if they're far away (in another country, say). We want an adverb, 'distantly', so that the sentence describes how closely those under investigation are financially related to the Mafia, not how far away they are. You can be 'related distantly'; you cannot be 'related distant'.
B and D also use adverbs, though because 'related' has been changed to 'relationships' in B and D, they would both need adjectives ('distant'). In E, it is unclear whether 'distant' refers back to 'relation' or to 'anyone' (and E has other problems).
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The problem with (A) is that to modify the verb 'related', you must use an adverb. The word 'distant' is an adjective. (B) To modify 'relationship' (a noun) we need 'distant' (an adjective) and not 'distantly' (an adverb). (C) This choice correctly modifies 'related' with 'distantly'. Choice (D) commits the same error that (B) does, in other words, it uses an adverb, 'distantly', to modify 'relationship'. (E) 'who' should be used instead of 'that'.