The answer should be (D).
In (A), 'doing it without' is awkward usage.
In (B), 'did it without' is grammatically incorrect.
In (E), the passive voice is used, and it is unclear who did without the timber and nails - the settlers or the homes
In (C), the usage is more awkward ('making them while not having') as compared to (D).
(D) is the best choice because it is in active voice, the settlers (subject) can clearly be seen to be acting on the timber and nails (object), and 'making do without' clearly conveys the meaning without the awkwardness in (C).
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