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NVAMediumJEE 2025Applications of P&C

JEE Mathematics 2025 Question with Solution

The number of 66-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

Correct answer:15

Step-by-step solution

Standard Method

Given: The letters of the word "MATHS" are available, and a 66-letter word is to be formed such that any letter that appears must appear at least twice.

Find: The total number of such 66-letter words.

From the solution:

  1. Identify the letters in the word "MATHS" and count how many times each can be repeated to form a 66-letter word.
  2. Use the counting principle to find the total number of valid words where each letter appears at least twice.
  3. The total number of such words is calculated as 1515.

Therefore, the required number of words is 1515.

Common mistakes

  • A common mistake is to allow a letter to appear only once. This violates the condition that any letter used in the word must appear at least twice. Instead, count only those arrangements in which every chosen letter occurs at least 22 times.

  • Another mistake is to treat all 55 letters of "MATHS" as if they must be used. The condition does not require every letter to appear; it only restricts the frequency of the letters that do appear. So the counting should be based on valid repetition patterns for a 66-letter word.

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