Question

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:

Hint:

When counting the number of functions that map to specific values, consider the number of choices for each element in the domain and apply the product rule for counting. In this case, choosing the one element to map to 1 gives us the total number of functions.

Answer 1

Determine the count of functions from the set \( \{1, 2, \dots, 100\} \) to the set \( \{0, 1\} \) where precisely one element from the domain \( \{1, 2, \dots, 100\} \) maps to 1, and all other elements map to 0.
- The selection of the single element from \( \{1, 2, \dots, 98\} \) that maps to 1 yields 98 possibilities. 
- Consequently, the remaining 99 elements in the set \( \{1, 2, \dots, 100\} \) must map to 0. The total number of such functions is therefore \( 98^{99} \). 
Final Answer: \( 98^{99} \).