Question

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ‘M’, is :

• A. 14950
• B. 6084
• C. 4356
• D. 5148

Hint:

The key to this problem is recognizing the positions of the letters before and after 'M' in the alphabet.

Answer 1

The Correct Option is D

To find the number of ways to choose and arrange five letters such that the middle one is 'M', we approach the problem by first recognizing the position of 'M'. It is the third letter in the sequence when arranged in alphabetical order. Thus, we need to select two letters from the alphabet that come before 'M' and two after it. Let's go through the steps:

1. Select letters before 'M': There are 12 letters (A to L) before 'M'. We choose 2 out of these 12. The number of combinations is given by the binomial coefficient: $\left(\genfrac{}{}{0ex}{}{12}{2}\right)$ which equals 66.
2. Select letters after 'M': There are 13 letters (N to Z) after 'M'. We choose 2 out of these 13, which gives us: $\left(\genfrac{}{}{0ex}{}{13}{2}\right)$ which equals 78.
3. Calculate total combinations: Since the selections before and after 'M' are independent of each other, we multiply the number of ways to select letters: 66 × 78 = 5148.

Therefore, the total number of ways to arrange five letters alphabetically with 'M' in the middle is 5148.

Answer 2

There are 26 letters in the English alphabet (A to Z).

We are choosing 5 letters such that when they are arranged in alphabetical order, the middle letter is ‘M’, i.e., the third letter.

Step 1: Fix the middle letter as ‘M’

Once we fix 'M' as the middle letter, we need to choose:

2 letters from the letters before ‘M’ (i.e., A to L → total 12 letters)

2 letters from the letters after ‘M’ (i.e., N to Z → total 13 letters)

Step 2: Choose the 2 letters before and after ‘M’

The number of ways to choose 2 letters from 12 before M is:
\( \binom{12}{2} = 66 \)

The number of ways to choose 2 letters from 13 after M is:
\( \binom{13}{2} = 78 \)

Step 3: Multiply the combinations

Since the letters must be in alphabetical order, we don't need to arrange them — each unique set leads to only one valid arrangement. So:

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

Final Answer:

Option 4: 5148