Originally posted by: pinion9
1) Yes, we have to assume we start somewhere.
2) What else would they be besides point locations?
3) It doesn't matter if the road is windy or straight. He is finding permutations of visitation, not distance.
4) We are NOT assuming that. We are actually assuming that the graph is completely connected and that ANY city can reach any other city.
5) Fair assumption. It still wouldn't make a difference because we are not asked about distance.
Please look up the travelling salesman problem.
Lets simplify the problem. Lets look at 2 cities. How many possible routes are there between the cities?
1) Yes, we have to assume we start somewhere. But the posters here have all seemed to assume we aren't starting in any of the 25 cities. My question is "Why are we assuming that"? This one assumption varies the answer substantially and we should write the "why" down with each answer so that people can understand the answer.
Lets assume we are in city #1 and assume that there is always only one road from our current location to the next. To go from city #1 to #2, there is one route. Thus, the answer is 1.
Lets assume we are out of both cities and assume that there is always only one road from our current location to the next. To get to city #1, there is one road. To get to from city #1 to city #2 there is one road. Or we could go first to city #2 and then to city #1. So there are two possible routes (starting->#1->#2 or starting ->#2->#1). The answer is two.
This one assumption in the most simple case doubles the answer. It is a major assumption.
2) Name one city in the world which is a point location. They all have at least some area under them. Need I go further (it ties in with comment 3 below)?
3) It does matter how the roads are. Don't confuse this question with the travelling salesman question. In real life there aren't just one road from point A to point B. There are long windy roads, and there are short direct roads. In reality, you have a very large (non-infinite, but still probably incomprehendible) number of roads to take from city #1 to city #2. To calculate the possible number of routes, we have to include all roads. Heck, this problem is quite difficult to solve even if you only have two cities. How many possible roads are between LA and NYC? The answer is NOT one road. Thus, how many routes are between LA and NYC. I don't know, but it certainly isn't just 1 possible route.
Of course if we assume we travel as the bird flies, then there is just one road between each city. So while, you might not actually travel straight, the effect is the exact same as if you assumed we travel as the bird flies.
Yes, I do realize calarification of the OP's problem would normally lead to this assumption. But as it is written it can be interpretted differently.
4) This ties in with comment (3).
5) Yes it would make a difference. Suppose city #9 annexed in a strange way as it developed such that it completely surrounds city #4. Now suppose we are in city #4 and want to do this problem. From city #4 we can ONLY travel to city #9. This cuts the possible number of routes down signficantly because we cannot travel from #4 to #25 or any other city but #9.
And to conclude, I think you failed to see my attempt at humor and you turned my post into a serious post.
LOL 
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