Let $n\in N$ $n\geq2$ and the set $X$ with $n+1$ elements. The ordered sequences $(a_{1}, a_{2},\ldots,a_{n})$ and $(b_{1},b_{2},\ldots b_{n})$ of distinct elements of $X$ are said to be $\textit{separated}$ if there exists $i\neq j$ such that $a_{i}=b_{j}$. Determine the maximal number of ordered sequences of $n$ elements from $X$ such that any two of them are $\textit{separated}$.
Note: ordered means that, for example $(1,2,3)\neq(2,3,1)$.
To determine the maximal number of ordered sequences of \( n \) elements from \( X \) such that any two of them are separated, we will follow the steps outlined in the solution and provide detailed explanations.

1. **Extend the Sequences:**
   - Given a set \( X \) with \( n+1 \) elements, consider an ordered sequence of \( n \) elements from \( X \). 
   - We can uniquely extend each sequence by adding the missing element in the \( (n+1)^{\text{th}} \) position. This transforms the problem into considering permutations of \( n+1 \) elements, denoted as \( \mathcal{S}_{n+1} \).

2. **Graph Construction:**
   - Construct a graph \( G = (V, E) \) where the vertices \( V \) are the elements of \( \mathcal{S}_{n+1} \) (i.e., all permutations of \( n+1 \) elements).
   - An edge exists between two vertices (permutations) \( \sigma_1 \) and \( \sigma_2 \) if and only if the sequences are **not** separated. This means there is no pair \( i \neq j \) such that \( a_i = b_j \).

3. **Characterizing Edges:**
   - For an edge \( \sigma_1 \sigma_2 \) to exist, there must be a unique \( 1 \leq i \leq n \) such that \( \sigma_2 = \tau(i, n+1) \circ \sigma_1 \), where \( \tau(i, n+1) \) is the transposition swapping the elements at positions \( i \) and \( n+1 \).

4. **Cycle Analysis:**
   - Consider a cycle \( \gamma \) in the graph \( G \). Traversing the cycle involves applying a sequence of transpositions.
   - Since each transposition has an odd signature, the length \( \lambda(\gamma) \) of the cycle must be even (as the product of an even number of odd permutations is even).
   - This implies that \( G \) contains no odd cycles, making \( G \) a bipartite graph.

5. **Bipartite Graph Properties:**
   - In a bipartite graph, the vertex set can be divided into two disjoint sets such that no two vertices within the same set are adjacent.
   - Each class of the bipartite graph \( G \) represents a set of pairwise separated sequences. The total number of vertices is \( (n+1)! \), so each class has at least \( \frac{1}{2} (n+1)! \) vertices.

6. **Maximal Set of Separated Sequences:**
   - Assume one class has more than \( \frac{1}{2} (n+1)! \) elements. By the pigeonhole principle, there would be at least two sequences with the same \( (n-1) \)-prefix, which cannot be separated, leading to a contradiction.
   - Therefore, the maximal number of ordered sequences such that any two are separated is exactly \( \frac{1}{2} (n+1)! \).

\[
\boxed{\frac{1}{2} (n+1)!}
\]