Problem of the Week-1: Election 2008

The Knox County Board of Commissioners, which has 20 members, recently had to elect a President. There were 3 candidates A (Alice), B (Bob), and C (Carol); on each ballot the 3 candidates were to be listed in order of preference, with no abstentions. It was found that 11 members (a majority ?) preferred A over B (thus the other 9 preferred B over A). Similarly it was found that 12 members preferred C over A. Given these results, it was suggested that B should withdraw to enable a runoff election between A and C. However, B protested, and it was then found that 14 members preferred B over C! The board has not yet recovered from resulting confusion. Given that every possible order of A,B,C appeared on at least one ballot, how many board members voted for B as their fist choice? How about A and C?

Justify your answer, show your solution.

Posted: 8/28/08

Submit your answers (by e-mail or hard copy) by 4 pm on 9/7/08 to Noah Aydin.

Mathematics Dept.