Let $X$ be a set of $n + 1$ elements, $n\geq 2$. Ordered $n$-tuples $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ formed from distinct elements of $X$ are called[i] disjoint [/i]if there exist distinct indices $1\leq i \neq j\leq n$ such that $a_i = b_j$. Find the maximal number of pairwise disjoint $n$-tuples.
1. **Understanding the Problem:**
   We need to find the maximal number of pairwise disjoint \( n \)-tuples formed from a set \( X \) of \( n+1 \) elements, where \( n \geq 2 \). Two \( n \)-tuples \((a_1, \ldots, a_n)\) and \((b_1, \ldots, b_n)\) are disjoint if there exist distinct indices \( 1 \leq i \neq j \leq n \) such that \( a_i = b_j \).

2. **Extending to Permutations:**
   Consider extending the \( n \)-tuples to permutations of the set \( [n+1] \). A permutation of \( [n+1] \) is a sequence that includes all elements of \( [n+1] \) exactly once. 

3. **Characterizing Non-Disjoint Permutations:**
   Two permutations \( \sigma_1 \) and \( \sigma_2 \) of \( [n+1] \) are not disjoint if there exists some \( i \in [n] \) such that \( \sigma_2 = (i, n+1) \sigma_1 \), where \( (a, b) \) denotes the transposition that swaps \( a \) and \( b \).

4. **Counting Permutations:**
   The total number of permutations of \( [n+1] \) is \( (n+1)! \). We can divide these permutations into even and odd permutations. By definition, an even permutation is one that can be expressed as an even number of transpositions, and an odd permutation is one that can be expressed as an odd number of transpositions.

5. **Bipartite Graph Construction:**
   Construct a bipartite graph where one part consists of the even permutations and the other part consists of the odd permutations. Draw an edge between two permutations if and only if they are not disjoint.

6. **Graph Properties and Hall's Theorem:**
   This bipartite graph is \( n \)-regular, meaning each vertex has exactly \( n \) edges. By Hall's Marriage Theorem, a bipartite graph with equal parts and each vertex having the same degree has a perfect matching. A perfect matching is a set of edges that matches each vertex in one part to exactly one vertex in the other part.

7. **Selecting Permutations:**
   Since the graph has a perfect matching, we can select at most one permutation from each edge of the matching. Therefore, we can select at most \( \frac{(n+1)!}{2} \) permutations.

8. **Conclusion:**
   The maximal number of pairwise disjoint \( n \)-tuples is \( \boxed{\frac{(n+1)!}{2}} \).