JEE Mathematics 2025 Question with Solution

Given: $f: \{1,2,\dots,100\} \to \{0,1\}$ and exactly one of the positive integers less than or equal to $98$ is assigned the value $1$.

Find: The number of such functions.

From the solution, exactly one element among $1,2,\dots,98$ must be mapped to $1$.

Choose that element in

$$\binom{98}{1} = 98$$

ways.

For the remaining $97$ elements among $1$ to $98$, the value must be $0$.

For $99$ and $100$, each can independently be assigned either $0$ or $1$, so the number of possibilities is

$$2^2 = 4$$

Therefore, the total number of functions is

$$98 \times 4 = 392$$

Hence, the required number of functions is $392$.

Given: the solution contains two approaches.

Find: The correct numerical answer.

In Approach Solution - 1, the written reasoning first says there are $98$ choices and that the remaining values are fixed to $0$, which would give $98$. However, the conclusion there states $392$, so that approach is internally inconsistent.

In Approach Solution - 2, the working explicitly states that exactly one among $1$ to $98$ is mapped to $1$, while the values at $99$ and $100$ are free.

Thus,

$$\binom{98}{1} = 98$$

and

$$2^2 = 4$$

So,

$$98 \times 4 = 392$$

Since this is the complete and consistent working shown in the solution, the answer is taken as $392$.