2010 AIME II Problems/Problem 8
Let be the number of ordered pairs of nonempty sets and that have the following properties:
- The number of elements of is not an element of ,
- The number of elements of is not an element of .
Let us partition the set into numbers in and numbers in ,
Since must be in and must be in (, we cannot partition into two sets of 6 because needs to end up somewhere, or either).
We have ways of picking the numbers to be in .
So the answer is .
Note: We have ways of picking the numbers to be in because there are numbers in and since is already a term in the set we simply have to choose another numbers from the numbers that are available.
Regardless of the size of (ignoring the case when ), must not be in and must be in .
There are remaining elements whose placements have yet to be determined. Note that the actual value of does not matter; there is always necessary element, forbidden element, and other elements that need to be distributed. There are places to put each of these elements, for possibilities.
However, there is the edge case of is forced not the be in either set, so we must subtract the cases where and have size .
Thus, our answer is
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