Permutations & Combinations
Choosing 5 Letters with Fixed Middle Letter
nta_pyq_2025_apr
Grade 11
Question:
The number of ways to choose 5 letters from the English alphabet in increasing alphabetical order such that the middle (3rd) letter is M is
Step-by-Step Solution
Key Concept: Since M must be the 3rd of 5 letters in ascending order, independently choose 2 letters from the 12 before M (A–L) and 2 letters from the 13 after M (N–Z).
M is the 13th letter of the alphabet.
Letters before M: A–L (12 letters). Letters after M: N–Z (13 letters).
Choose 2 from A–L and 2 from N–Z:
$$\binom{12}{2}\times\binom{13}{2}=66\times78=5148.$$
Correct Answer: 1