The Navajo people adapted to life in captivity in New Mexico in the se

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