Within 30 days after appointment and qualification of a member, the board shall meet and elect from among its members a chairperson and such other officers as the board may provide for by rule. This organization shall continue until the appointment of a new member or upon the resignation of an existing board officer. The office of secretary and treasurer may be held by one member. A majority of the members of the board who have been approved by the governor and council shall constitute a quorum for the transaction of business.