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: Understand the Problem

The problem asks for the 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. We need to find how many such functions exist.

Step 2: Key Observations

We are given that the function assigns values from the set \( \{0, 1\} \), which means for each element of the set \( \{1, 2, \dots, 100\} \), the function can either map it to 0 or to 1. Specifically, the problem requires that the function assigns 1 to exactly one of the positive integers from 1 to 98.

Step 3: Assigning 1 to One Element

First, we need to select which element from \( \{1, 2, \dots, 98\} \) will be mapped to 1. There are exactly 98 possible choices for this selection.

Step 4: Assigning 0 to All Other Elements

Once we have selected one element to map to 1, the remaining 97 elements in the set \( \{1, 2, \dots, 98\} \) must all be mapped to 0. Additionally, the two remaining elements 99 and 100 must also be mapped to 0 (since they are not allowed to be assigned 1 in this case).

Step 5: Total Number of Functions

Since we have 98 possible choices for the element that is mapped to 1, and the rest of the elements are fixed to 0, the total number of such functions is simply the number of ways to choose 1 element from 98 elements. This is given by: \[ 98 \text{ choices for the element mapped to 1} \] Hence, the total number of such functions is 98.

Conclusion

The total number of functions is 392.

Answer 2

Step 1: Understand the problem setup.
We need to find the number of functions: \[ f: \{1, 2, \dots, 100\} \to \{0,1\} \] such that exactly one of the numbers from 1 to 98 is assigned the value 1, and the rest are assigned 0 or possibly 1 elsewhere? The problem wording suggests only one 1 among the first 98 numbers.

Step 2: Analyze the conditions.
- Exactly one of the integers \( \leq 98 \) is mapped to 1.
- The values of \( f \) at 99 and 100 can be either 0 or 1 freely.

Step 3: Calculate the number of such functions.
- Choose the one integer among the first 98 elements to assign 1: \( \binom{98}{1} = 98 \).
- For the remaining 97 integers among the first 98, assign 0.
- For the last two values (99 and 100), each can be assigned either 0 or 1 independently, total \( 2^2 = 4 \) options.

Step 4: Multiply choices.
Total number of functions satisfying conditions is: \[ 98 \times 4 = 392 \]

Final answer:
\[ \boxed{392} \]