(a) The responsibility for paying the expenses for transporting, examining, and caring for clients, which expenses are not provided for under Chapter 9 of this title, relating to the payment of costs of care of persons admitted to state institutions under the department, shall be in the following order:

(1) The client or his estate;

(2) Persons legally obligated or legally responsible for the support of the client;

(3) The county of the client's legal residence;

(4) The department, when the General Assembly appropriates funds for such purpose.

(b) The department is authorized to issue rules and regulations governing the provisions of this Code section as it relates to the department.