UPSC CSE Prelims 2022
A, B and C are three places such that there are three different roads from A to B. four different roads from B to C and three different roads from A to C. In how many different ways can one travel from A to C using these roads?
EXPLANATION
Correct Option (3)
To ascertain the total number of distinct ways to travel from place A to place C, two primary categories of routes must be considered, as they are mutually exclusive:
- Direct Route from A to C: There are 3 distinct roads provided for direct travel between A and C.
- Indirect Route from A to C via B: This route involves sequential travel from A to B, followed by travel from B to C.
- Number of roads from A to B = 3.
- Number of roads from B to C = 4.
- According to the fundamental principle of counting (multiplication principle for sequential events), the total number of ways to travel from A to C via B is the product of the number of ways for each segment: 3 × 4 = 12 ways.
The total number of different ways to travel from A to C is the sum of the ways for the direct route and the indirect route, as these represent all possible and distinct travel paths.
Total ways = (Ways via Direct Route) + (Ways via Indirect Route)
Total ways = 3 + 12 = 15.
Incorrect Options:
Option 1 (10): This value is incorrect. It might result from an erroneous summation of the number of roads (e.g., 3 roads A-B + 4 roads B-C + 3 roads A-C = 10), which does not correctly apply the principles of counting for sequential or alternative routes.
Option 2 (13): This value is incorrect. It suggests a miscalculation in either the indirect route or the final summation. For instance, if the indirect route was incorrectly determined as 10 (e.g., 3+4+3) and then added to the direct route (3+10=13), or other similar misapplications of counting principles.
Option 4 (36): This value is incorrect. It would result from an erroneous multiplication of all given road counts (3 × 4 × 3 = 36), which misinterprets the problem's structure as a single sequence of three events rather than two distinct, mutually exclusive travel paths.