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

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: \( 98^{99} \).