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 determine the number of arrangements of five letters where the third letter is 'M', we first fix the position of 'M'. This means we must select two letters preceding 'M' and two letters succeeding 'M' from the alphabet. The process is as follows:

1. Selection of preceding letters: There are 12 letters from 'A' to 'L' that precede 'M'. The number of ways to choose 2 of these is calculated using combinations: \( \binom{12}{2} \), resulting in 66.
2. Selection of succeeding letters: There are 13 letters from 'N' to 'Z' that succeed 'M'. The number of ways to choose 2 of these is calculated using combinations: \( \binom{13}{2} \), resulting in 78.
3. Total combinations calculation: As the selections are independent, the total number of combinations is the product of the individual selection counts: 66 × 78 = 5148.

Thus, there are 5148 distinct ways to arrange five letters alphabetically with 'M' as the third letter.