If there is a contradiction between something the Rambam wrote in the Mishneh Torah and his responsum letters, which is considered his final ruling? Is there a general rule or is it a case by case?
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Great question. It's a problem that is not limited to the Rambam, you can find the same phenomenon among other poskim.
For example, the Maharil "contradicts" himself regarding when one removes shoes before the Ninth of Av. Does one take them off at home, or before Ma'ariv? That depends whether you read the responsa or the "Customs" (מנהגי מהרי"ל). See here: http://hebrewbooks.org/pdfpager.aspx?req=8859&st=&pgnum=111
Here is a series of shiurim that covers the issue:
R. David Yosef in the introduction to the P'er Hador edition of Rambam's responsa quotes many who value the MT over the responsa, including the Rashba, in addition to citing some who hold the opposite, such as the Radvaz.
It's generally understood that the Mishna Torah was Rambam's last -- and therefore greatest -- work. We assume that he changed his mind over time from his responsa-writing period to when he wrote the Yad HaChazaka (i.e. Mishna Torah).
(Though much ink has been spilled about his responsum allowing conversion to Islam rather than death, vs. the Yad's prohibition.)