A combinatorics problem on connecting the cities

In summary, the conversation discusses the concept of connecting fifteen cities in a way where each city has one road leading to five other cities. It is mentioned that there will be a total of 75 road ends, which is not an even number and thus making it impossible to construct the roads as each road has two ends. A question is asked for clarification on the impossibility, to which the explanation is given that each city will have 5 road ends, leading to the total of 75.
  • #1
Vineeth T
31
0
Fifteen cities are planned to be connected in such a way that each city has precisely one road leading to each of five other cities.How many such roads are to be constructed?

This question was asked in a talent test conducted in our school.
 
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  • #2
5*15=75 which is not an even number. So it is impossible.
 
  • #3
Rogerio said:
5*15=75 which is not an even number. So it is impossible.
Hi! Rogerio
Can you explain me clearly why is it impossible?
 
  • #4
Assuming that a road always leads from one city to another city, each city has 5 road ends connected to this city, so we have a total of exactly 75 road ends.
Every road has 2 ends - the total number of road ends has to be even. 75 is not even.
 
  • #5


I find this combinatorics problem interesting and challenging. The number of roads to be constructed in this scenario can be calculated by using the formula for combinations. Since each city has to be connected to five other cities, we can consider this as selecting 5 cities out of 15 cities. Therefore, using the combination formula, the number of roads to be constructed will be 15C5 = 3003. This means that a total of 3003 roads will be needed to connect all 15 cities in the given way. However, it is important to note that this is the minimum number of roads required, as there could be multiple ways to connect the cities while still meeting the given criteria. Overall, this problem showcases the importance of combinatorics in solving real-world problems and highlights the complexity of city planning.
 

1. What is combinatorics?

Combinatorics is a branch of mathematics that deals with counting and arranging objects or elements in a finite set. It involves the study of combinations, permutations, and other related concepts.

2. What is the "connecting the cities" problem?

The "connecting the cities" problem is a combinatorics problem that involves finding the number of possible routes or paths that can be taken to travel between a set of cities. The main objective is to determine the total number of ways to connect all the cities without repeating any route.

3. What factors affect the complexity of the "connecting the cities" problem?

The complexity of the "connecting the cities" problem depends on the number of cities, the available modes of transportation, and the restrictions or rules that govern the routes. The more cities there are and the more restrictions there are, the more complex the problem becomes.

4. What strategies can be used to solve the "connecting the cities" problem?

There are several strategies that can be used to solve the "connecting the cities" problem, such as using graph theory, dynamic programming, or brute force. Each strategy has its own advantages and disadvantages, and the most suitable approach may depend on the specific problem and its constraints.

5. What real-life applications does the "connecting the cities" problem have?

The "connecting the cities" problem has various real-life applications, such as in transportation planning, network optimization, and logistics management. It can also be used to model the movement of goods, people, or information in a system, making it an important tool in decision-making and problem-solving.

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