Mich. Comp. Laws § 201.35: Vacancies in office; county officers; appointment, term.

Sec. 5.

When a vacancy shall occur in an elective or appointive county office, it shall be filled in the following manner:

If the vacancy shall be in the office of county clerk or prosecuting attorney it shall be filled by appointment by the judge or judges of that judicial circuit and the person appointed shall hold office for the remainder of the unexpired term.

If the vacancy shall be in any other county office, either elective or appointive, the judge of probate, the county clerk and the prosecuting attorney shall appoint some suitable person to fill such vacancy and the person so appointed shall hold such office for the remainder of the unexpired term.