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

Answer 1

Answer: (D)

5148

Answer 2

The problem asks us to find the total number of ways to choose five English alphabets such that when arranged in alphabetical order, the middle letter is 'M'.

Let the five chosen letters be $L_1, L_2, L_3, L_4, L_5$. Since they are arranged in alphabetical order, we have the condition $L_1 < L_2 < L_3 < L_4 < L_5$. We are given that the middle letter is 'M', which means $L_3 = 'M'$.

Now, we need to select two letters that come before 'M' and two letters that come after 'M'.

1. Letters before 'M': The letters in the English alphabet that come before 'M' are A, B, C, D, E, F, G, H, I, J, K, L. There are 12 such letters. We need to choose 2 distinct letters from these 12 letters for positions $L_1$ and $L_2$. Since they will be arranged in alphabetical order, the order of selection does not matter; choosing any two letters automatically places them in the correct ascending order. The number of ways to choose 2 letters from 12 is given by the combination formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. Number of ways to choose $L_1$ and $L_2$ = $\binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 6 \times 11 = 66$.

2. Letters after 'M': The letters in the English alphabet that come after 'M' are N, O, P, Q, R, S, T, U, V, W, X, Y, Z. There are 13 such letters (26 total letters - 13 letters from A to M = 13 letters). We need to choose 2 distinct letters from these 13 letters for positions $L_4$ and $L_5$. Similar to the previous step, the order of selection does not matter. The number of ways to choose 2 letters from 13 is given by the combination formula: Number of ways to choose $L_4$ and $L_5$ = $\binom{13}{2} = \frac{13 \times 12}{2 \times 1} = 13 \times 6 = 78$.

To find the total number of ways, we multiply the number of ways to choose the letters before 'M' and the number of ways to choose the letters after 'M', as these are independent choices.

Total number of ways = (Number of ways to choose $L_1, L_2$) $\times$ (Number of ways to choose $L_4, L_5$) Total number of ways = $66 \times 78 = 5148$

Thus, there are 5148 ways in which five English alphabets can be chosen and arranged in alphabetical order such that the middle letter is 'M'.