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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.