Question:
Four cities are connected by a road network as shown in the figure (a triangle with vertices and a centroid-like point connected to all three vertices). In how many ways can you start from any city and come back to it without travelling on the same road more than once?
Four cities are connected by a road network as shown in the figure (a triangle with vertices and a centroid-like point connected to all three vertices). In how many ways can you start from any city and come back to it without travelling on the same road more than once?
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Every city is joined to the other three, and each city has 3 roads. Count the closed routes from one fixed city.
Updated On: Nov 5, 2026
- 8
- 12
- 16
- 20
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Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Read the figure:
The three corners of the triangle are joined to each other and each is also joined to the point inside. So each of the 4 cities is joined directly to the other 3: there are 6 roads in all.
Step 2: Shape of a closed route:
Every city has only 3 roads. Passing through a city uses 2 roads, so a route cannot pass through a city twice without repeating a road. So a closed route is a simple loop.
Step 3: Count from one city, say A:
Loops through 3 cities: choose the two other cities in order, \(3 \times 2 = 6\) routes. Loops through all 4 cities: visit the other three in some order, \(3! = 6\) routes. Total \(6 + 6 = 12\). By symmetry it is the same from any city.
Final Answer:
The correct option is (2): 12.
The three corners of the triangle are joined to each other and each is also joined to the point inside. So each of the 4 cities is joined directly to the other 3: there are 6 roads in all.
Step 2: Shape of a closed route:
Every city has only 3 roads. Passing through a city uses 2 roads, so a route cannot pass through a city twice without repeating a road. So a closed route is a simple loop.
Step 3: Count from one city, say A:
Loops through 3 cities: choose the two other cities in order, \(3 \times 2 = 6\) routes. Loops through all 4 cities: visit the other three in some order, \(3! = 6\) routes. Total \(6 + 6 = 12\). By symmetry it is the same from any city.
Final Answer:
The correct option is (2): 12.
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