Permutations and CombinationsMCQMTP Dec 22 - Series IQuestion 1660 of 249
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How many words can be formed with the letters of the word 'ORIENTAL' So that A and E always occupy odd places:

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option c — 8640

All Options:

  • A540
  • B8460
  • C8640
  • D8450

Detailed Solution & Explanation

The word is ORIENTAL, which contains 8 distinct letters: {O,R,I,E,N,T,A,L}\displaystyle \{O, R, I, E, N, T, A, L\}.
Let's separate them into vowels and consonants:
- Vowels: {O,I,E,A}\displaystyle \{O, I, E, A\} (4 vowels)
- Consonants: {R,N,T,L}\displaystyle \{R, N, T, L\} (4 consonants)
The 8 letter positions are 1, 2, 3, 4, 5, 6, 7, 8.
- Odd places: 1, 3, 5, 7 (4 places)
- Even places: 2, 4, 6, 8 (4 places)
We want the two specific vowels A\displaystyle A and E\displaystyle E to always occupy odd places.
1. Select 2 odd positions out of the 4 available odd positions and arrange A\displaystyle A and E\displaystyle E in them:
Ways=4P2=4×3=12 waysWays=4P2=4\displaystyle \text{Ways} = ^4P_2 = 4 ×\displaystyle \times 3 = 12  ways\displaystyle \text{ ways}
2. The remaining 6 positions must be filled by the remaining 6 letters (vowels O,I\displaystyle O, I and consonants $R, N, T, L$):
Ways=6!=720 waysWays=6\displaystyle \text{Ways} = 6! = 720  ways\displaystyle \text{ ways}
Total number of arrangements is:
Total=4P2×6!=12×720=8640Total=4P2\displaystyle \text{Total} = ^4P_2 ×\displaystyle \times 6! = 12 ×\displaystyle \times 720 = 8640
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.

Related Comparison Tables

Permutations vs Combinations: The Quantitative Aptitude Guide

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