The [i]Olimpia[/i] country is formed by $n$ islands. The most populated one is called [i]Panacenter[/i], and every island has a different number of inhabitants. We want to build bridges between these islands, which we'll be able to travel in both directions, under the following conditions:

a) No pair of islands is joined by more than one bridge.

b) Using the bridges we can reach every island from Panacenter.

c) If we want to travel from Panacenter to every other island, in such a way that we use each bridge at most once, the number of inhabitants of the islands we visit is strictly decreasing.

Determine the number of ways we can build the bridges.
1. **Understanding the Problem:**
   - We have \( n \) islands, each with a unique population.
   - The most populated island is called Panacenter.
   - We need to build bridges under the following conditions:
     a) No pair of islands is joined by more than one bridge.
     b) Every island must be reachable from Panacenter.
     c) When traveling from Panacenter to any other island, the number of inhabitants must strictly decrease.

2. **Graph Representation:**
   - Represent the islands as vertices in a graph.
   - Represent the bridges as edges in the graph.
   - Condition (b) implies the graph must be connected.
   - Condition (c) implies that the graph must be a tree (a connected acyclic graph) with Panacenter as the root.

3. **Tree Structure:**
   - Since the population strictly decreases as we move away from Panacenter, each island \( I_k \) (where \( k > 1 \)) can only be connected to one of the islands \( I_1, I_2, \ldots, I_{k-1} \).
   - This ensures that there are no cycles and the population condition is maintained.

4. **Counting the Ways to Build Bridges:**
   - For \( I_2 \), it can only be connected to \( I_1 \) (Panacenter).
   - For \( I_3 \), it can be connected to either \( I_1 \) or \( I_2 \).
   - For \( I_4 \), it can be connected to \( I_1, I_2, \) or \( I_3 \).
   - Generally, \( I_k \) can be connected to any of the \( k-1 \) islands with a higher population.

5. **Total Number of Ways:**
   - The number of ways to connect \( I_2 \) is 1.
   - The number of ways to connect \( I_3 \) is 2.
   - The number of ways to connect \( I_4 \) is 3.
   - And so on, up to \( I_n \), which can be connected in \( n-1 \) ways.
   - Therefore, the total number of ways to build the bridges is the product of these choices:
     \[
     (n-1)!
     \]

Conclusion:
\[
\boxed{(n-1)!}
\]