Set theory question

[Bobmuhthol]

I'm having a lot of trouble with a question that I shouldn't have a lot of trouble with. Suppose there are 2 sets, A and B, with n and m elements, respectively. How many elements are in the set {C | B included in C included in A}. Obviously the maximum value is 2^n (and the answer should therefore be able to be expressed as 2^n - expression), which is only possible if B is empty, and the minimum is 1, which is only possible if A = B, but I can't figure out a formula that will actually determine how many subsets of A do or do not contain B. I've tried using Pascal's triangle but I don't see a pattern, other than that you always include the last 1 and discard the first terms until you get to the number of elements in B.

Any help would be appreciated because I am going insane.

[Cephalopod]

http://i.imgur.com/n092i.jpg

Sorry, I tried my hardest.

[Bobmuhthol]

My inability to make an exhaustive list was throwing me off. The answer is 2^(n-m) because the entire set B can be considered as a requisite group of elements in each subset, and the subsets can contain additional elements from A, effectively creating a new Pascal of equal or lesser order.

I am embarrassed at how long this took me.