The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______.
Answer: 1405
Solution
The word MATHS has 5 distinct letters. In a six-letter arrangement where every letter used appears at least twice, the number of distinct letters used can only be 1, 2 or 3, since these are the only ways to partition 6 positions into groups of size at least 2.
Using exactly one letter: choose the letter in 5 ways, giving 5 words.
Using exactly two letters: choose the pair in ways. The six positions split as or . For a fixed pair, the split can be assigned in 2 ways (either letter takes the 2), each giving arrangements, and the split gives arrangements. That is words per pair, so words.
Using exactly three letters: choose the triple in ways; the only valid split is , giving arrangements per triple, so words.
Total: .
Using exactly one letter: choose the letter in 5 ways, giving 5 words.
Using exactly two letters: choose the pair in ways. The six positions split as or . For a fixed pair, the split can be assigned in 2 ways (either letter takes the 2), each giving arrangements, and the split gives arrangements. That is words per pair, so words.
Using exactly three letters: choose the triple in ways; the only valid split is , giving arrangements per triple, so words.
Total: .
JEE Main · 2025 · 29 Jan 2025 · Shift 1 · Integer answer