Let $X$ be a finite set. Suppose $R$ is some proper subset of X. Let $S$ be the set of subsets of $X$ of a given size; for example, $S$ could be the set of three-membered subsets of $X$. Is there any way to calculate how many members of $S$ have $R$ as a subset?
[edited this question due to some unclarities in the formulation]
You rightly mentioned that if $|R|>|s|,s\in S$, then the answer is $0$. On the other hand, if $|s|\ge|R|$, you can count the number of subsets of $X$ of size $|s|$ with $R$ as a subset. For that, note that $|R|$ elements are already fixed; they are the elements of $R$. You can select the remaining $|s|-|R|$ elements from the remaining $|X|-|R|$ elements of $X$ in $\displaystyle\binom{|X|-|R|}{|s|-|R|}$ ways, which is the answer.