late 15c., "easy to do," from French facile "easy," from Latin facilis "easy to do," of persons, "pliant, courteous, yielding," from facere "to do" (from PIE root *dhe- "to set, put"). Usually now with depreciatory implication. Of persons, "easily led," from 1510s.
late 14c., "want of easiness, that quality which makes something laborious or perplexing," from Anglo-French difficulté and directly from Latin difficultatem (nominative difficultas) "difficulty, distress, poverty," from difficilis "hard," from dis- "not, away from" (see dis-) + facilis "easy to do," from facere "to do" (from PIE root *dhe- "to set, put"). From 1610s as "that which is difficult." Related: Difficulties.
early 15c., "gentleness, lightness," from Old French facilité "easiness, ease," from Latin facilitatem (nominative facilitas) "easiness, ease, fluency, willingness," from facilis "easy to do," from facere "to do" (from PIE root *dhe- "to set, put"). First in a medical book:
If it be nede forto smyte [the head] wiþ a malle, be it done with esynez or facilite [transl. Guy de Chauliac's "Grande Chirurgie"]
Its sense in English expanded to "opportunity" (1510s), to "aptitude, ease, quality of being easily done" (1530s). Meaning "place for doing something" which makes the word so beloved of journalists and fuzzy writers, first recorded 1872, via notion of "physical means by which (something) can be easily done."
Academic sense "branch of knowledge" (late 14c.) was in Old French and probably was the earliest in English (it is attested in Anglo-Latin from late 12c.), on notion of "ability in knowledge" or "body of persons on whom are conferred specific professional powers." Originally each department was a faculty; the use in reference to the whole teaching staff of an entire college dates from 1767. Related: Facultative. The Latin words facultas and facilitas "were originally different forms of the same word; the latter, owing to its more obvious relationship to the adj., retained the primary sense of 'easiness', which the former had ceased to have before the classical period." [OED]