Fundamental Counting Principle

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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?

RankGuru · Accepted Answer

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.

Fundamental Counting Principle

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