Permutations and CombinationsPYQ Sept 25Question 4427 of 183
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Different words are made with rearrangement of letters of the word "TROPICAL" in a way that the vowels occupy odd places when counted from left. How many such words are there?

Options

A720
B1440
C2880
D2160
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Correct Answer

✅ Option c — 2880

All Options:

  • A720
  • B1440
  • C2880
  • D2160

Detailed Solution & Explanation

The word is "TROPICAL". Let us analyze the letters: - Total number of letters = 8 (all letters are distinct) - Vowels: O, I, A (3 vowels) - Consonants: T, R, P, C, L (5 consonants)
The word has 8 positions, numbered 1 to 8 from left to right. - Odd positions: 1, 3, 5, 7 (Total of 4 odd positions) - Even positions: 2, 4, 6, 8 (Total of 4 even positions)
According to the condition, the vowels must occupy odd positions. - We have 3 vowels and 4 odd positions. The number of ways to place the 3 vowels in the 4 odd positions is: 4P3=4×3×2=24 ways^4P_3 = 4 \times 3 \times 2 = 24\text{ ways}
- The remaining 5 positions (1 odd position and 4 even positions) will be occupied by the 5 consonants. The number of ways to arrange the 5 consonants in these 5 positions is: 5!=120 ways5! = 120\text{ ways}
The total number of words that can be formed is: Total words=4P3×5!=24×120=2880 words\text{Total words} = ^4P_3 \times 5! = 24 \times 120 = 2880\text{ words}
Hence, **Option C** 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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