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 :

Solution

Maths sir · 3 steps

Fix M in the middle, then pick two letters before it and two after it. The order takes care of itself.

1. Order is fixed. Once five letters are chosen, alphabetical order allows only one arrangement. So you are only counting selections.
2. Split the alphabet. For M to be third, exactly 2 letters must come before M and 2 after it. There are 12 letters before M (A to L) and 13 letters after M (N to Z).

   ¹²C₂ × ¹³C₂ = 66 × 78 = 5148

3. So. The number of ways is 5148, option (b).

The idea to keep. When chosen items must sit in a fixed order, arrangements disappear and only selection remains. Fix the given item, then choose freely from each side of it.

Answer from the NTA final answer key. Solution by PrepWiser.