Question 24

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______________

Correct Answer: 392

Solution

Functions f:{1,...,100}→{0,1} assigning 1 to exactly one integer ≤ 98.
Choose which of 1-98 gets value 1: 98 ways. For positions 99,100: each can be 0 or 1, but we need exactly one value of 1 among 1-98. Positions 99 and 100 can be 0 or 1 freely: 4 choices.

Total = 98 × 4 = 392.
The answer is 392.

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