The treasurer shall be appointed by the board and serve at its pleasure. At least 90 days before the date of a general district election wherein the offices of assessor and tax collector will be subject to election, the board may determine that these officers shall henceforth be appointed by the board to serve at its pleasure, or, if a board has adopted the alternative provisions of Part 7.5 (commencing with Section 37200) of this division, it may by resolution eliminate these district offices. A person appointed pursuant to this section need not be a resident or elector of the district. Any person holding an elective office affected by this section may serve the remainder of the current term of office. The offices of treasurer, assessor and tax collector, or any combination thereof, may be held by the same person.