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:

Answer: 392

Step-by-step solution

Step 1: Understand the function definition

The function $f$ maps elements from the set $A = \{1, 2, \ldots, 100\}$ to the set $B = \{0, 1\}$ . This means for each element in the domain, the function can assign either 0 or 1.

Step 2: Identify the constraint

The problem states that exactly one of the positive integers less than or equal to 98 is assigned the value 1. This means we need to choose one element from the set $\{1, 2, \ldots, 98\}$ to be mapped to 1.

Step 3: Determine choices for elements up to 98

Since exactly one integer from $\{1, 2, \ldots, 98\}$ must be assigned 1, there are 98 possible choices for which integer gets the value 1. All other $98-1=97$ integers in this range must be assigned 0.

Step 4: Determine choices for remaining elements

The constraint only applies to integers less than or equal to 98. For the remaining integers, 99 and 100, there is no restriction. Each of these two integers can be assigned either 0 or 1 independently.

Step 5: Calculate total number of functions

We multiply the number of choices for each part of the domain. There are 98 ways to choose the integer less than or equal to 98 that gets 1. There are 2 choices for $f(99)$ (0 or 1) and 2 choices for $f(100)$ (0 or 1).

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