Embedded subsets

Determine the number:

Variant
Of ordered pairs \((A, B)\), where \(A \subseteq B \subseteq \{1, 2, …, n\}\).
Hint
Try to describe the choice of subsets \(A\) and \(B\) using an appropriate mapping.
Solution
We introduce a characteristic function \(\chi_{A,B}\), where \( \chi_{A,B}(x)= \begin{cases} 0 & x\in \{1,…,n\}\setminus A \\ 1 & x\in A \setminus B \\ 2 & x\in B \end{cases} \)

Every choice of \((A,B)\) gives a unique characteristic function, and conversely.

The number of pairs \((A,B)\) equals the number of functions from \(\{1,…,n\}\) to \(\{0{,}1,2\}\).
Answer
The number of possible choices \((A,B)\) is \(3^n\).
Variant
Of ordered quadruples \((A, B, C, D)\), where \(A \subseteq B \subseteq D \subseteq \{1, 2, …, n\}\) and also \(A \subseteq C \subseteq D\).
Solution
We introduce a characteristic function \(\chi_{A,B,C,D}\), where \( \chi_{A,B,C,D}(x)= \begin{cases} 0 & x\in \{1,…,n\}\setminus D \\ 1 & x\in D \setminus (B \cup C) \\ 2 & x\in B \setminus C \\ 3 & x\in C \setminus B \\ 4 & x\in (B\cup C) \setminus A \\ 5 & x\in A \end{cases} \)

Each choice \((A,B,C,D)\) give a unique characteristic function, and conversely.

The number of quadruples \((A,B,C,D)\) equals the number of functions from \(\{1,…,n\}\) to \(\{0,…,5\}\).
Answer
The number of possible choices \((A,B,C,D)\) is \(6^n\)