Counting subsets of disjoint sets
The set, , contains numbers: .
The set, , contains any subset of with cardinality :
are some pairwise disjoint sets that satisfy .
The set, , contains any subset of with cardinality , whose elements are pairwise disjoint. With , it looks like this:
I need to figure out:
1. How many elements of contains at least one element of
2. How many elements of contains no element of and elements of
Since any element of is element of exactly one of the above described sets, the sum is the same
as the cardinality of . With being the pochhammer symbol, is:
I need to look through and count some stuff of to obtain what's needed to determine the result,
but I have no freaking idea about what and how to use it.. any ideas please?
Re: Counting subsets of disjoint sets
Maybe use Inclusion/Exclusion? Let . Then, the number of elements of that contain at least one element of would be the number of elements of containing plus the number containing plus ... plus the number containing , minus the number containing exactly two of the subsets, plus/minus ...