Mich. Comp. Laws § 168.209

Sec. 209. If a vacancy occurs in an elective or appointive county office, it shall be filled in the following manner: (1) If the vacancy is in the office of county clerk or prosecuting attorney, it shall be filled by appointment by the judge or judges of that judicial circuit. (2) If the vacancy is in any other county office, the presiding or senior judge of probate, the county clerk, and the prosecuting attorney shall appoint a suitable person to fill the vacancy. (3) A person appointed shall take and subscribe to the oath as provided in section 1 of article XI of the state constitution of 1963, give bond in the manner required by law, and hold office for the remainder of the unexpired term and until a successor is elected and qualified. However, if the vacancy occurs more than 7 days before the nominating petition filing deadline as provided in section 193 for the general November election that is not the general November election at which a successor in office would be elected if there were no vacancy, the person appointed shall hold office only until a successor is elected at the next general November election in the manner provided by law and qualifies for office. The successor shall hold the office for the remainder of the unexpired term.