Exercise 24

The Wohascum County Board of Commissioners, which has 20 members, recently had to elect a President. There were three candidates ($A$, $B$, and $C$); on each ballot the three 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 the resulting confusion. Given that every possible order of $A$, $B$, $C$ appeared on at least one ballot, how many members voted for $B$ as their first choice?