IrMO 2011 Problem 5

In the mathematical talent show called "The $X^2$ -factor" contestants are scored by a a panel of $8$ judges. Each judge awards a score of $0$ ('fail'), $X$ ('pass'), or $X^2$ ('pass with distinction'). Three of the contestants were Ann, Barbara and David. Ann was awarded the same score as Barbara by exactly $4$ of the judges. David declares that he obtained different scores to Ann from at least $4$ of the judges, and also that he obtained different scores to Barbara from at least $4$ judges.

In how many ways could scores have been allocated to David, assuming he is telling the truth?