Number of functions f : {1,2,...,100} → {0,1}, that assign 1 to exactly one of the positive integers

Q: Wait, why do we split the domain into two subsets $S_1$ and $S_2$?

Great question! In combinatorics, we often use a 'divide and conquer' strategy. The problem imposes a specific constraint on the first 98 elements, while the last 2 are completely unrestricted. By splitting the domain, we isolate the constrained part from the free part, making the calculation much cleaner and less prone to errors.

Q: Why is the number of ways for $S_2$ equal to $2^2$?

Remember that for each element in the domain, we have a choice in the codomain. Since $S_2 = \{99, 100\}$ has no constraints, element 99 can map to either 0 or 1 (2 choices), and element 100 can also map to either 0 or 1 (2 choices). Since these choices are independent, we multiply them: $2 \times 2 = 4$.

Q: What if the question said 'at least one' instead of 'exactly one'?

That would change the game entirely! If it were 'at least one', it would be easier to calculate the total number of functions ($2^{98}$) and subtract the case where *none* of the elements map to 1. Always look for keywords like 'exactly', 'at least', or 'at most'—they are the keys to choosing your counting method.

Question

Number of functions $f:\{1,2,...,100\}\rightarrow\{0,1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to

Sigma Percentile · Accepted Answer

### The Art of Counting: A Journey Through Functions

Welcome, fellow traveler in the world of mathematics! Today, we are going to unravel a beautiful problem in combinatorics. It is not just about finding a number; it is about understanding the structure of functions.

Imagine you are standing before a set of 100 items, and you have to assign each of them a value of either 0 or 1. There is a constraint: exactly one element from the set $\{1, 2, \dots, 98\}$ must map to 1. Let us break this down together.

### Analyzing the Setup

When a problem seems large, the first instinct of a mathematician is to simplify. We have a domain $A = \{1, 2, 3, \dots, 100\}$ and a codomain $B = \{0, 1\}$.

The constraint is local; it only cares about the first 98 integers. So, let us partition our domain into two distinct subsets:

1. $S_1 = \{1, 2, \dots, 98\}$
2. $S_2 = \{99, 100\}$

By separating the constrained elements from the free ones, we turn one complex problem into two manageable ones. This is the essence of combinatorial thinking.

### The Special One in $S_1$

Now, let us focus on $S_1$. The problem demands that exactly one element in this set maps to 1.

Think of this as choosing a 'special' element. If we have 98 elements, the number of ways to choose one to be the special one is given by the combination formula:

$$^{98}C_1 = 98$$

This gives us 98 ways to pick the element that maps to 1. Once that element is chosen, all other 97 elements in $S_1$ are forced to map to 0. They have no choice, so the number of ways to map $S_1$ is exactly 98.

### The Freedom of $S_2$

Now, look at $S_2 = \{99, 100\}$. The problem is silent about these two, meaning they are free agents.

Element 99 can map to 0 or 1, providing 2 choices. Element 100 can also map to 0 or 1, providing another 2 choices.

Since these choices are independent, the total number of ways to map $S_2$ is:

$$2 	imes 2 = 2^2 = 4$$

### Final Calculation

We have solved the two parts of our puzzle. We have 98 ways to handle the constrained set $S_1$ and 4 ways to handle the free set $S_2$.

According to the fundamental principle of counting, if we have two independent tasks, we multiply the number of ways to perform each. Therefore, the total number of valid functions is:

$$98 	imes 4 = 392$$

And there you have it! By breaking the problem into logical pieces, we have navigated through the constraints and arrived at the solution with clarity and confidence. The final answer is **392**.

Animated Solution for Mathematics - Permutations and Combinations: Number of functions $f : {1, 2, ..., 100} \to {0, 1}$ , that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to

Enter Numerical Value:

Visualized Solution

Defining the Sets

Splitting the Domain

Constraint on Subset $S_{1}$

Calculating Ways for $S_{1}$

Analyzing Subset $S_{2}$

Calculating Ways for $S_{2}$

Total Number of Functions

The Sigma Insight: Fundamental Principle of Counting

The Art of Counting

Analyzing the Setup

The Special One in $S_{1}$

The Freedom of $S_{2}$

Final Calculation

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