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Simple answer: you don't get to claim amounts over $4000. The maximum amount for the American Opportunity Credit (AOC) is also $4000. That gets you an (up to) $2500 credit. The AOC is the most generous credit.
That said, if the education is related to her current job and/or she has a 529 plan, there MAY be some additional tax breaks available. Otherwise, $4000 is all that is allowed.