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The correct answer is DNA fingerprinting because it is the most reliable method for establishing paternity.
Option 1, protein analysis, is insufficient because proteins are determined by genes, but not all variations or similarities in proteins can definitively prove or disprove a genetic relationship.
Option 2, chromosome counting, is also not a reliable method for establishing paternity. While it can help in identifying certain genetic disorders and conditions, it cannot determine relationships because all humans have the same number of chromosomes.
Option 3, quantitative analysis of DNA, may be able to provide some insight, but quantity of DNA does not necessarily equate with genetic relatedness or paternity. For instance, two completely unrelated people could have similar amounts of DNA.
Option 4, DNA fingerprinting, on the other hand, is a technique that examines the unique aspects of an individual`s DNA - their "genetic fingerprint". By comparing these fingerprints between a child and a potential father, scientists can identify if there is a match, thus proving paternity. This technique is currently the most reliable and commonly used method for these types of determinations.