Fundamental Counting Principle
A student must travel from city A to city C passing through city B, with 4 roads from A to B and 5 roads from B to C; the number of distinct routes from A to C equals which value?
Select the correct option:
Solution
20
The fundamental principle of counting states that if one task can be done in m ways and a following independent task in n ways, the combined task can be done in m times n ways, the bedrock of JEE Advanced counting. Travelling from A to B can be done in 4 ways, and for each such choice, travelling from B to C can be done in 5 ways. Since the two legs are independent and sequential, the total number of routes is 4 × 5 = 20. Option 9 wrongly adds the choices, which would apply only if the legs were mutually exclusive alternatives. Option 45 multiplies by an incorrect factor. Option 1 ignores the multiplicity of roads. Hence there are 20 distinct routes. Plausibility check: listing the routes as ordered pairs (road to B, road to C) yields a grid of 4 rows and 5 columns, totalling 20 cells, exactly matching the multiplication principle.
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About This Question
- Subject
- mathematics
- Chapter
- permutations and combinations
- Topic
- fundamental counting principle
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
20
The fundamental principle of counting states that if one task can be done in m ways and a following independent task in n ways, the combined task can be done in m times n ways, the bedrock of JEE Advanced counting. Travelling from A to B can be done in 4 ways, and for each such choice, travelling from B to C can be done in 5 ways. Since the two legs are independent and sequential, the total number of routes is 4 × 5 = 20. Option 9 wrongly adds the choices, which would apply only if the legs were mutually exclusive alternatives. Option 45 multiplies by an incorrect factor. Option 1 ignores the multiplicity of roads. Hence there are 20 distinct routes. Plausibility check: listing the routes as ordered pairs (road to B, road to C) yields a grid of 4 rows and 5 columns, totalling 20 cells, exactly matching the multiplication principle.
This easy difficulty mathematics question is from the chapter permutations and combinations, covering the topic of fundamental counting principle. It appeared in the 2025 exam.
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