Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{5,7,8\}$.
To solve the problem, we need to find the number of pairs \(\{A, B\}\) such that \(A \subseteq X\), \(B \subseteq X\), \(A \neq B\), and \(A \cap B = \{5, 7, 8\}\).

1. **Identify the fixed elements in \(A\) and \(B\)**:
   Since \(A \cap B = \{5, 7, 8\}\), both sets \(A\) and \(B\) must contain the elements 5, 7, and 8. Therefore, we can consider the remaining elements in \(X\), which are \(\{1, 2, 3, 4, 6, 9, 10\}\).

2. **Determine the subsets of the remaining elements**:
   The remaining elements are \(\{1, 2, 3, 4, 6, 9, 10\}\), which has 7 elements. Each of these elements can either be in \(A\), in \(B\), or in neither \(A\) nor \(B\). However, they cannot be in both \(A\) and \(B\) simultaneously because \(A \cap B = \{5, 7, 8\}\).

3. **Calculate the number of choices for each element**:
   For each of the 7 remaining elements, there are 3 choices:
   - The element is in \(A\) but not in \(B\).
   - The element is in \(B\) but not in \(A\).
   - The element is in neither \(A\) nor \(B\).

4. **Compute the total number of pairs**:
   Since there are 7 elements and each has 3 choices, the total number of pairs \(\{A, B\}\) is:
   \[
   3^7
   \]

5. **Subtract the invalid case where \(A = B\)**:
   The only invalid case is when \(A = B\). In this case, each of the 7 elements must be in neither \(A\) nor \(B\). There is exactly 1 such case.

6. **Final calculation**:
   Therefore, the number of valid pairs \(\{A, B\}\) is:
   \[
   3^7 - 1
   \]

\[
3^7 = 2187
\]
\[
2187 - 1 = 2186
\]

The final answer is \(\boxed{2186}\).