JEE Mathematics 2025 Question with Solution

Given: Five letters are chosen from the English alphabet and arranged in alphabetical order.

Find: The number of ways such that the middle letter is $M$.

Since the letters are arranged in alphabetical order, $M$ must be the third letter. So we must choose:

• two letters from $A$ to $L$
• two letters from $N$ to $Z$

There are $12$ letters before $M$ and $13$ letters after $M$.

The number of ways to choose two letters before $M$ is

$$\binom{12}{2} = 66$$

The number of ways to choose two letters after $M$ is

$$\binom{13}{2} = 78$$

Since these choices are independent,

$$\text{Total ways} = \binom{12}{2} \times \binom{13}{2} = 66 \times 78 = 5148$$

Therefore, the total number of ways is $5148$. The correct option is D.

Given: Five letters are to be chosen and written in alphabetical order.

Find: The number of selections for which the middle letter is $M$.

If $M$ is the middle letter, then exactly two selected letters must be smaller than $M$ and exactly two selected letters must be greater than $M$.

Letters before $M$ are $A$ to $L$, so there are $12$ such letters. Letters after $M$ are $N$ to $Z$, so there are $13$ such letters.

Choose the two smaller letters:

$$\binom{12}{2}$$

Choose the two greater letters:

$$\binom{13}{2}$$

Because the final arrangement is already fixed by alphabetical order, each valid choice of five letters gives exactly one arrangement. Thus,

$$\binom{12}{2} \cdot \binom{13}{2} = 66 \cdot 78 = 5148$$

Therefore, the required number of ways is $5148$, so the correct option is D.