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:

Step 1: Understand the problem constraints

We need to choose 5 letters from the 26 English alphabets and arrange them in alphabetical order. The key constraint is that the middle letter must be 'M'.

Step 2: Determine letters before 'M'

Since the letters are arranged in alphabetical order and 'M' is the middle letter (the 3rd letter), the two letters before 'M' must be chosen from the letters that come before 'M' in the alphabet. There are 12 such letters (A to L).

Step 3: Choose 2 letters before 'M'

We need to choose 2 distinct letters from the 12 letters available before 'M'. The order of selection does not matter because they will be arranged alphabetically anyway. This is a combination problem, calculated as $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.

Step 4: Determine letters after 'M'

Similarly, the two letters after 'M' must be chosen from the letters that come after 'M' in the alphabet. There are 13 such letters (N to Z).

Step 5: Choose 2 letters after 'M'

We need to choose 2 distinct letters from the 13 letters available after 'M'. Again, the order of selection does not matter.

Step 6: Calculate total number of ways

To find the total number of ways, we multiply the number of ways to choose the letters before 'M' by the number of ways to choose the letters after 'M'. This is because these choices are independent events.

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