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

Step 1: Problem Statement

Determine the count of functions \( f: \{1, 2, \dots, 100\} \to \{0, 1\} \) where exactly one element from the set \( \{1, 2, \dots, 98\} \) is mapped to 1.

Step 2: Conditions

Each input from \( \{1, 2, \dots, 100\} \) maps to either 0 or 1. The constraint is that precisely one integer in the range \( [1, 98] \) is mapped to 1.

Step 3: Selecting the Element Mapped to 1

Choose one element from the set \( \{1, 2, \dots, 98\} \) to be assigned the value 1. There are 98 distinct options for this selection.

Step 4: Assigning Values to Remaining Elements

The selected element is mapped to 1. All other 97 elements within \( \{1, 2, \dots, 98\} \) must be mapped to 0. Furthermore, elements 99 and 100 must also be mapped to 0, as the condition specifies 1 is assigned only within the first 98 integers.

Step 5: Calculating Total Functions

The total number of functions is determined by the number of ways to choose the single element that maps to 1 from the 98 available options. All other mappings are then fixed to 0. \[ 98 \text{ choices for the element mapped to 1} \] Therefore, there are 98 such functions.

Conclusion

The total number of functions is 98.