The Navajo people adapted to life in captivity in New Mexico in the second half of the nineteenth century, bringing a distinctive weaving style with them, through which was maintained both a livelihood and a way of preserving their traditional culture.
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(A) a distinctive weaving style with them, through which was maintained both a livelihood and
"through which" modifies "distinctive weaving style". Hence they must be closer to each other but there is another modifier "with them" in between those two. Doesn't look good.
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(B) a distinctive weaving style with them, and through which maintaining both a livelihood and
Same problem as in option A. The modifiers are distant but they must be closer to each other.
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(C) with them a distinctive weaving style, through which they maintained both a livelihood and
All the modifiers are closer to the phrase they modify. "they" clearly refers back to Navajo people and correct use of idiom "both X and Y". Looks good.
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(D) with them a distinctive weaving style, through which
maintains both a livelihood and
Meaning error. "maintains both X and Y" Who maintains is not clear.
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(E) with them a distinctive weaving style, and maintaining through it both a livelihood and
"maintaining through it" the verb tense doesn't sound correct.
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Hence option C