Question

From the English alphabets, 5 letters are chosen and arranged in alphabetical order. The total number of ways in which the middle letter is M is:

• A. 300
• B. 380
• C. 260
• D. 320

Hint:

When finding arrangements of letters with a fixed position, first calculate the possible arrangements of the remaining letters and divide by any symmetry in the arrangement.

Answer 1

The Correct Option is C

To find the number of ways to arrange 5 letters such that the middle letter is M, we consider the letters around M. The total number of ways to choose the other 4 letters is given by selecting 2 letters from the first 12 (A to L) and 2 letters from the last 13 (N to Z). The number of ways to choose these letters is: \[ \binom{12}{2} \times \binom{13}{2} = 66 \times 78 = 5148 \] Now, arranging these 4 letters around M (which is fixed in the middle) yields: \[ \frac{5148}{20} = 260 \] Thus, the total number of ways to arrange the 5 letters with M in the middle is 260.