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 : (1) 14950 (2) 6084 (3) 4356 (4) 5148

Key Concepts Needed

1. Combination vs. Permutation

   • Combination counts selections when order does not matter.
   • Permutation counts arrangements when order does matter.

2. Alphabetical Order Fixes the Arrangement
   Once we promise to write the letters in A→Z order, any chosen set of 5 distinct letters gets only one valid arrangement.

3. Middle Position Logic
   In a 5-letter alphabetical list (positions 1-5), the 3rd position is the middle.
   If that position must be M, then:

   • Positions 1 & 2 must be letters earlier than M.
   • Positions 4 & 5 must be letters later than M.

4. Counting Choices

   • Letters before M: A … L ⇒ 12 letters.
   • Letters after M: N … Z ⇒ 13 letters.
   • Choose 2 out of 12 and 2 out of 13 using the combination formula $\binom{n}{r}=\frac{n!}{r!\,(n-r)!}$.

5. Total Ways
   Multiply the two independent selections: $\binom{12}{2}\times\binom{13}{2}$.

Imagine a Long Alphabet Rope

1. Think of the English alphabet as a rope with 26 beads: A, B, C … Z.
2. You have to pick 5 beads and then lay them exactly in alphabetical order.
3. The teacher says, “Make sure the middle bead (the 3rd one) is the letter M!”
4. So you only worry about choosing:
   • 2 beads that come before M (from A to L)
   • 2 beads that come after M (from N to Z)
5. Once you pick those beads, the order is already fixed (alphabetical), so there is only one way to arrange them.
6. The whole game now becomes a simple count-how-many-ways-to-choose problem!

Step-by-Step Solution

Step 1: Identify letters available on each side of M
Letters before M: A … L → 12 letters.
Letters after M: N … Z → 13 letters.

Step 2: Choose 2 letters that come before M

Step 3: Choose 2 letters that come after M

Step 4: Multiply the independent choices

Step 5: Final Answer
The required number of ways is 5148.
Hence, option (4) 5148 is correct.