In 1981, a young and ambitious Donald Trump sat down with federal agents and discussed his calculation in entering the mob-infested world of Atlantic City casinos.

He acknowledged it might tarnish the reputation his family built through traditional real-estate development in New York. He was aware a business partner in the New Jersey beach town knew people who might be unsavory. A Federal Bureau of Investigation agent advised Mr. Trump there were easier ways he could invest, said an FBI memo recounting the discussion.

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