Problem 1252

1. **Count the number of ordered triples $(A, B, C)$ such that $A \subseteq B \subseteq C$:**

For each element $x \in X$, there are 4 possibilities:
- $x$ belongs to all three sets $A$, $B$, and $C$.
- $x$ belongs to $B$ and $C$ but not to $A$.
- $x$ belongs to $C$ only.
- $x$ belongs to none of the sets.

Since there are $n$ elements in $X$, the total number of such triples is:
$$4^n$$

2. **Count the number of ordered triples $(A, B, C)$ such that $A \subseteq B = C$:**

For each element $x \in X$, there are 3 possibilities:
- $x$ belongs to both $A$ and $B = C$.
- $x$ belongs to $B = C$ but not to $A$.
- $x$ belongs to none of the sets.

Since there are $n$ elements in $X$, the total number of such triples is:
$$3^n$$

3. **Subtract the number of triples where $B = C$ from the total number of triples:**

The number of ordered triples $(A, B, C)$ such that $A \subseteq B \subseteq C$ and $B \neq C$ is:
$$4^n - 3^n$$

The final answer is $\boxed{4^n - 3^n}$