Word 'REGULATION' is arranged without repetition. Find the probability that the vowels come at odd places.
Select Your Answer (Keyboard: A, B, C, D)
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Permutations and CombinationsMCQMTP Dec 23 - Series IIQuestion 1674 of 249
All Questions Correct Answer
✅ Option a — 1
All Options:
- A1
- B1
- C252
- D144
Detailed Solution & Explanation
We want to find the probability that the vowels occupy only the odd places when the letters of the word "REGULATION" are arranged randomly without repetition.
First, let us analyze the letters in the word "REGULATION":
- Total number of letters = 10.
- Vowels: (5 vowels).
- Consonants: (5 consonants).
In a 10-letter arrangement, the positions are numbered $1, 2, 3, \dots, 10\displaystyle .<br>- Odd places: $1, 3, 5, 7, 9 (exactly 5 positions).
- Even places: $2, 4, 6, 8, 10\displaystyle (exactly 5 positions).<br><br>For the vowels to occupy the odd positions:<br>- The 5 vowels must be arranged in the 5 odd places. The number of ways to do this is $5! = 120.
- The 5 consonants must be arranged in the remaining 5 even places. The number of ways to do this is $5! = 120P1/252$ is represented as Option A.
Hence, **Option A** is the correct answer.
First, let us analyze the letters in the word "REGULATION":
- Total number of letters = 10.
- Vowels: (5 vowels).
- Consonants: (5 consonants).
In a 10-letter arrangement, the positions are numbered $1, 2, 3, \dots, 10\displaystyle .<br>- Odd places: $1, 3, 5, 7, 9 (exactly 5 positions).
- Even places: $2, 4, 6, 8, 10\displaystyle (exactly 5 positions).<br><br>For the vowels to occupy the odd positions:<br>- The 5 vowels must be arranged in the 5 odd places. The number of ways to do this is $5! = 120.
- The 5 consonants must be arranged in the remaining 5 even places. The number of ways to do this is $5! = 120P1/252$ is represented as Option A.
Hence, **Option A** 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 SyllabusExam 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.
Key Concepts to Understand
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