(1) In each county containing a city of the first class there shall be a bipartisan board known as the "Board for (name of city) and (....) County Children's Home." (2) The board shall be composed of fourteen (14) members, at least two (2) of whom shall be women, appointed jointly by the mayor of the city and the county judge/executive of the county. Each member shall be at least twenty-five (25) years of age and a bona fide resident of the county. The members shall be appointed in equal proportion from the two (2) political parties casting the highest and next highest number of votes at the last preceding general election in the county.
(3) The members of the board shall be divided into four (4) groups, of which two (2) groups shall consist of three (3) members each and two (2) groups shall consist of four (4) members each. Members shall serve for four (4) years each, and the terms of the members of any one (1) group shall expire at the end of a year which is different from the year in which the terms of the members of any other group expire. Members originally appointed during the year following the effective date hereof may be appointed for such terms as may be necessary to effectuate the rotation provided for in the preceding sentence.
(4) Vacancies in the board, whether occurring during the term or at the close of the term of a member, shall be filled in the manner provided in subsection (2) of this section, within thirty (30) days after the vacancy occurs, and if the mayor and the county judge/executive fail to make the appointment within thirty (30) days, the members of the board may make the appointment. Whenever any vacancy occurs, the board shall immediately certify the fact of the vacancy and the reason therefor to the county judge/executive and the mayor. All members shall serve without compensation.