Sets, Relations & Functions
Counting Functions
nta_pyq_2025_apr
Grade 11
Question:
Number of functions $f: \{1, 2, \ldots, 100\} \to \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ___
Step-by-Step Solution
Key Concept: Choose which element of $\{1,\ldots,98\}$ maps to 1 (98 ways), while the remaining elements of $\{1,\ldots,98\}$ must map to 0. Elements 99 and 100 can independently map to 0 or 1.
Choose 1 element from $\{1,\ldots,98\}$ to map to 1: 98 ways. Element 99 maps to 0 or 1: 2 ways. Element 100 maps to 0 or 1: 2 ways. Total $= 98 \times 2 \times 2 = 392$.
Correct Answer: 392