Number Of Functions

Question

The total number of functions that can be defined from a set with 5 elements to a set with 3 elements, with no further restriction, equals which value?

RankGuru · Accepted Answer

A function assigns to each domain element exactly one codomain element, and these choices are independent, a counting principle central to JEE Advanced. Each of the 5 domain elements can be mapped to any of the 3 codomain elements, giving 3 choices per element. By the multiplication principle, the total number of functions is 3^5 = 243. The exponent is the domain size and the base is the codomain size, the canonical form (codomain)^(domain). Option 125 = 5^3 reverses base and exponent. Option 15 = 5·3 wrongly adds choices instead of multiplying. Option 8 confuses this with a subset count of a 3-set. Hence the total is 3^5 = 243. Plausibility check: the count must exceed the number of injective functions, which is zero here since the domain is larger than the codomain so no injection exists, and 243 comfortably accommodates all mappings including many non-injective ones, confirming the formula.

Number Of Functions

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