If sets $A$ and $B$ are subsets of the set $\{1, 2, \ldots, n\}$, with $|A| = k$ and $|B| = m$, where $k \leq m$, the task is to determine the total number of pairs $(A, B)$ where $A \subseteq B \subseteq \{1, 2, \ldots, n\}$.
For $m$ and $k$ fixed we get $\binom{m}{n}\binom{k}{m}$ pairs. When $k$ varies and $m$ is fixed we get $\binom{0}{n}\binom{m}{n} + \binom{1}{n}\binom{m-1}{n-1} + \ldots + \binom{m}{n}\binom{0}{n-m} = 2^m \binom{m}{n}$. pairs
I have solved the problem when $k$ and $m$ are fixed and when only $m$ is fixed and I encountered difficulties when both $m$ and $k$ vary. I would really like to know what would also happen when we have $X_1 \subseteq X_2\subseteq... \subseteq X_n\subseteq \{1, 2, \ldots, n\}$.