2010 AIME II Problem 8

Let N be the number of ordered pairs of nonempty sets \mathcal{A} and \mathcal{B} that have the following properties:

  • \mathcal{A} \cup \mathcal{B}=\{1,2,3,4,5,6,7,8,9,10,11,12\}
  • \mathcal{A} \cap \mathcal{B}=\emptyset,
  • The number of elements of \mathcal{A} is not an element of \mathcal{A},
  • The number of elements of \mathcal{B} is not an element of \mathcal{B}. Find N.