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 and Explanation
We need to find the number of functions from the set \( \{1, 2, \dots, 100\} \) to the set \( \{0, 1\} \), such that exactly one of the values in the domain \( \{1, 2, \dots, 100\} \) is mapped to 1, and all other values are mapped to 0.
- First, we select which element from \( \{1, 2, \dots, 98\} \) will be mapped to 1. There are 98 choices for this.
- Then, the remaining 99 elements in the set \( \{1, 2, \dots, 100\} \) must all be mapped to 0. Thus, the total number of functions is \( 98^{99} \).
Final Answer: \( 98^{99} \).
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