| Each of these sentences has a double negative. Our book says that barely is a negative word. Hasn't has the word not in it, so choice #1 is out. Same with choice #2, nothing is a negative, and so is barely. Choice #2 is out. In choice #3, can't is a negative (after all, it contains the word not. Hardly is negative, so choice #3 is out. The book tells us that we can correct a negative by changing one of the two negative words to a positive. In Choice #4, can't is a negative, but anything is a positive (positive form of nothing). So it's right.
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