Explanation:

Correct Answer is (c).The original sentence is erroneous.
Reason: The correct idiomatic expression is 'at arm’s length' and not 'at arming lenghts'.
At arm’s length (Idiom):
Meaning: Avoiding intimacy or close contact.
E.g.: John always keeps his friends at arm's length, so that no one gets close enough to hurt his feelings.
Hence ‘at arm’s length’ should be used in place of ‘at arming lengths’ to make the sentence grammatically correct.
Among the given choices, only option C replaces the given bold part most appropriately.
The sentence after replacement becomes:
Despite the political collusion, our suspicious chief minister has kept her comrades at arm’s length.Option C is hence the correct answer.