Let S={1,2,....,n}. The number of possible pairs of the form (A,B) with $A \subseteq B$ for subsets $A,B$ of $S$ is

Given: $S = \{1, 2, \ldots, n\}$
We need to find number of pairs $(A, B)$ such that $A \subseteq B \subseteq S$
Method: Element-wise analysis
For each element $i \in S$, there are three possibilities:
$i \notin A$ and $i \notin B$
$i \notin A$ and $i \in B$
$i \in A$ and $i \in B$
Note: The case $i \in A$ and $i \notin B$ is not possible since $A \subseteq B$
So each element independently has exactly $3$ choices
Since there are $n$ elements in $S$:
Total number of pairs $= \underbrace{3 \times 3 \times \cdots \times 3}_{n \text{ times}}$
$\therefore \boxed{3^n}$