The road network forms a graph. To start from a city and return to it without traversing the same road twice means finding elementary cycles (circuits) that pass through the starting city. Let's label the cities A, B, C, D. The figure likely shows a complete graph K4, where every city is connected to every other city by a road. If it's K4: - From city A, you can go to B, C, or D (3 options). - If you go A -> B, you can then go B -> C or B -> D (2 options). - If you go A -> B -> C, you must go C -> D to visit all cities or C -> A to return directly. Since you must return to the start and not use the same road twice, you need to complete a cycle. - Cycles of length 3 involving A: A-B-C-A, A-C-B-A, A-B- D-A, A-D-B-A, A-C-D-A, A-D-C-A. (6 ways starting from A). - Cycles of length 4 involving A: A-B- C-D-A, A-B-D-C-A, A-C-B-D-A, A-C-D-B-A, A- D-B-C-A, A-D-C-B-A. (6 ways starting from A). Total ways starting from A and returning to A = 6 (length 3) + 6 (length 4) = 12. Since the graph is symmetric (a complete graph), the number of ways is the same regardless of the starting city.
Walked East for 200 meters. Current position relative to start: 200m East. (200, 0) 2. Turned left (North) and walked 150 meters. Current position relative to previous: 150m North. Net displacement from start: 200m East, 150m North. (200, 150) 3. Turned right (East) and walked 250 meters. Current position relative to previous: 250m East. Net displacement from start: (200 + 250)m East, 150m North = 450m East, 150m North. (450, 150) 4. Turned right (South) and walked 150 meters. Current position relative to previous: 150m South. Net displacement from start: 450m East, (150 - 150)m North = 450m East, 0m North. (450, 0)The final position is 450 meters East and 0 meters North/ South from the starting point. The distance from the starting point (0,0) to the final point (450,0) is the straight-line distance. Distance = $\sqrt{(450-0)^2 + (0-0)^2} = \sqrt{450^2} = 450$ meters. 65. Four cities are connected by a road network as shown in the figure. In how many ways can one start from any city and come back to it without travelling on the same road more than once?
Asked in GPSC Prelims CSP 1 2019 · Topic: General Concept (Spatial Distribution and Patterns (Fundamental Idea))
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