Key Concepts and Formulas

• Functions: A function $f: A \rightarrow B$ assigns each element of set $A$ to exactly one element of set $B$.
• Combinations: The number of ways to choose $k$ objects from a set of $n$ distinct objects is given by the binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
• Counting Principle: If there are $n$ ways to do one thing and $m$ ways to do another, then there are $n \times m$ ways to do both.

Step-by-Step Solution

Step 1: Identify the constraint

We are looking for functions $f:\{1,2, \ldots, 100\} \rightarrow\{0,1\}$ such that exactly one of the integers from 1 to 98 is assigned the value 1. This is the key constraint of the problem. The remaining integers can be either 0 or 1.

Step 2: Choose the integer from 1 to 98 that is mapped to 1

We need to choose one integer from the set $\{1, 2, \ldots, 98\}$ to be mapped to 1. There are 98 choices for this.

Step 3: Consider the remaining integers from 1 to 98

Since exactly one of the integers from 1 to 98 must be mapped to 1, all the remaining 97 integers in the set $\{1, 2, \ldots, 98\}$ must be mapped to 0.

Step 4: Consider the integers 99 and 100

The integers 99 and 100 can each be mapped to either 0 or 1. This gives us $2$ choices for 99 and $2$ choices for 100. Since these choices are independent, we have $2 \times 2 = 4$ possibilities for the mappings of 99 and 100.

Step 5: Calculate the total number of functions

We have 98 ways to choose which integer from 1 to 98 is mapped to 1. For each of these choices, the other 97 integers from 1 to 98 must be mapped to 0. Then, we have 4 possibilities for mapping 99 and 100. Thus, the total number of functions is $98 \times 4 = 392$.

Common Mistakes & Tips

• Misinterpreting the constraint: A common mistake is to not fully understand the constraint that exactly one of the integers from 1 to 98 must be mapped to 1.
• Forgetting about 99 and 100: Remember that the function is defined on the set $\{1, 2, \ldots, 100\}$, so you must consider the mappings of 99 and 100.
• Double Counting: Be careful not to double-count the number of functions. Ensure each function is counted only once.

Summary

We are looking for the number of functions $f:\{1,2, \ldots, 100\} \rightarrow\{0,1\}$ such that exactly one integer from 1 to 98 is assigned the value 1. We first choose which of the 98 integers is mapped to 1. Then, the remaining integers from 1 to 98 are all mapped to 0. Finally, we consider the possible mappings of 99 and 100, which can each be 0 or 1. Multiplying these counts together gives the total number of functions. The total number of such functions is $98 \times 2 \times 2 = 392$.

Final Answer The final answer is \boxed{392}.