In electing members of the Board of Regents pursuant to article 13, section 3, of the Constitution of the state of Minnesota, and Territorial Laws 1851, chapter 3, section 5, one member of the Board of Regents of the university shall be a person who at the time of election to the board is a student who is enrolled in a degree program at the university. This person shall represent the state at large. Upon expiration of the term or in the event of a vacancy in the office, one position shall be filled by a person having the same qualifications.