Number of Combinations for Connecting N Villages with One-Way Roads

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Homework Help Overview

The problem involves determining the number of ways to connect N villages using one-way roads, ensuring that there is at least one route between any two villages. The subject area encompasses combinatorics and graph theory.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the initial attempts to find solutions for small values of N, with one noting a trivial solution for N=2 and another suggesting a recursive approach. Questions are raised regarding the clarity of the problem statement, particularly about the number of routes allowed between villages and any restrictions on road connections.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem and questioning the assumptions made in the original statement. Some guidance has been offered regarding potential approaches, such as recursion, but no consensus has been reached.

Contextual Notes

Participants are considering the implications of the problem's constraints, including the possibility of multiple routes between villages and limits on road connections, which remain undefined in the original question.

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Homework Statement


There are N villages which have to be connected using one-way roads in a manner that for every two villages there exists at least one route to travel from one to another.
How many different solutions are there?


Homework Equations


This fits pretty well not only into combinatorics but also in a graph theory.


The Attempt at a Solution


For N=2 there is only 1 trivial solution.
For N=3 there seems to be 15 solutions, but it's done by writing out the combinations and does not lead to general solution.
 
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I'm not proposing an answer, but I don't think you have well stated the question. Can there be more than one route that works for two cities? Is there some limit on how many roads can enter/leave a city? What are the restrictions?
 
Perhaps you should start out by trying to find some recursive solution, and then using mathematical induction to make it non-recursive. A lot of solutions to problems such as this in Graph Theory involve recursivity. (Is that a word?)
 
LCKurtz said:
I'm not proposing an answer, but I don't think you have well stated the question. Can there be more than one route that works for two cities? Is there some limit on how many roads can enter/leave a city? What are the restrictions?

I am guessing it should be no more than 1 route from any A to any B.
 
Whovian said:
recursivity. (Is that a word?)

The word is "recursion" :)
 

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