This is a question of misplaced modifiers. Many of the wrong answers can be eliminated by spotting the faulty modification. (E), however, is a problem of verb tense, not of modification. Let's take a close look:
a.) Faulty voting equipment, confusing ballots, voter error, and problems at polling places have been cited by a new study of the 2000 United Stated presidential
election, which estimated that they did not count 4 million to 6 million of the 100 million votes cast.
The election did not estimate, the study did. b.) Citing faulty voting equipment, confusing ballots, voter error, and problems at polling places, a new study of the 2000 United States presidential election has estimated that 4 million to 6 million of the 100 million votes cast were not counted.
ANSWERStudy is being correctly modified. The verb tense of present perfect mixed with simple past is correct. c.)
Citing faulty voting equipment, confusing ballots, voter error, and problems at polling places,
4 million to 6 million of the 100 million votes cast were not counted in the 2000 United States presidential election, a new study estimates.
Dangling modifier at the beginning. 'Citing...' must modify the study. d.)
A new study has cited faulty voting equipment, confusing ballots, voter error, and problems at polling places
in estimating that 4 million to 6 million of the 100 million votes that were cast had not been counted in the 2000 United States presidential election.
Here the 'in estimating' is stylistically off, as it is very far from the subject it applies to, 'a new study.' There are also some less obvious errors in verb tense. The votes cast should be in the past perfect tense, 'had not been cast' because this event happened before the counting.e.) A new study of the 2000 United States presidential election, citing faulty voting equipment, confusing ballots, voter error, and problems at polling places,
has estimated 4 million to 6 million votes
had not been counted of the 100 million votes cast.
'Has estimated' is mixing present perfect tenses with the past perfect, 'had not been.' Either use past and past perfect or present and present perfect. Hope that helps

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