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

JEE Main · 2025 · 24 Jan 2025 · Shift 2 · Permutations and Combinations

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 ____.

Answer: 392

A function

f : {1, 2, \dots, 100} \to {0, 1}

assigning 1 to exactly one element of

{1, \dots, 98}

is built in three independent steps.

Choose which single element of

{1, \dots, 98}

maps to 1: 98 ways. All the remaining elements of

{1, \dots, 98}

are then forced to map to 0 (only 1 way).

The elements 99 and 100 lie outside the restricted set, so each can independently map to 0 or 1:

2 \times 2 = 4

ways.

Multiplying these independent choices:

98 \times 1 \times 4 = 392.

JEE Main · 2025 · 24 Jan 2025 · Shift 2 · Integer answer