Permutations and CombinationsPYQ Sept 25Question 4424 of 183
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How many different words from the letters of the word MATHEMATICS can be formed so that all the vowels always come together in any word?

Options

A10080
B120960
C4989600
D20160
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Correct Answer

✅ Option b — 120960

All Options:

  • A10080
  • B120960
  • C4989600
  • D20160

Detailed Solution & Explanation

The given word is "MATHEMATICS". Let us analyze the letters: - Total number of letters = 11 - Frequency of letters: M (2), A (2), T (2), H (1), E (1), I (1), C (1), S (1) - The vowels are: A, E, A, I (Total vowels = 4, with A repeated 2 times) - The consonants are: M, T, H, M, T, C, S (Total consonants = 7, with M repeated 2 times and T repeated 2 times)
Since all the vowels must always come together, we treat the group of vowels (A, E, A, I) as a single entity/unit. Therefore, we need to arrange: - 7 consonants - 1 single vowel group unit This is a total of 7+1=8\displaystyle 7 + 1 = 8 units.
The number of ways to arrange these 8 units (where M is repeated 2 times and T is repeated 2 times) is: Arrangement of units=8!2!×2!=40,3204=10,080 ways\text{Arrangement of units} = \frac{8!}{2! \times 2!} = \frac{40,320}{4} = 10,080\text{ ways}
Now we must arrange the 4 vowels within their group unit (where A is repeated 2 times): Arrangement of vowels=4!2!=242=12 ways\text{Arrangement of vowels} = \frac{4!}{2!} = \frac{24}{2} = 12\text{ ways}
The total number of words that can be formed is the product of these two arrangements: Total words=10,080×12=1,20,960 words\text{Total words} = 10,080 \times 12 = 1,20,960\text{ words}
Hence, **Option B** is the correct answer.

About This Chapter: Permutations and Combinations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Factorials, Permutations, Combinations

This chapter deals with the fundamental principles of counting. It covers factorials, circular permutations, restricted permutations, combinations, and the differences between selecting items versus arranging them.

View Official ICAI Syllabus

Exam Strategy Tip

The most common mistake is confusing 'P' (Arrangement) with 'C' (Selection). If order matters (like opening a lock), use P. If order doesn't matter (like choosing a team), use C.

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