JEE Main 2025 Jan 24 Shift 2, Mathematics Q21: Introduction Of Functions

JEE Main2025Jan 24, Shift 2Mathematics

Number of functions $f : {1, 2, \dots, 100} \to {0, 1}$ , that assign $1$ to exactly one of the positive integers less than or equal to $98$ , is equal to ______.

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Solution

Split the domain into two parts:

Inputs $1, 2, \dots, 98$ (98 elements): the condition requires that exactly one of these receives the value $1$ , and all others receive $0$ .
Inputs $99, 100$ (2 elements): unrestricted — each can independently be mapped to $0$ or $1$ .

Number of ways to choose which of ${1, \dots, 98}$ gets mapped to $1$ : $(1 98) = 98$ .

Number of ways to assign values to $f (99)$ and $f (100)$ : $2 \times 2 = 4$ .

Total $= 98 \times 4 = 392$ .

Answer: 392

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JEE Main 2025 Jan 24 Shift 2, Mathematics Q21: Introduction Of Functions

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