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

For .

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 ...