Question:
The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is ___________
The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is ___________
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Be careful with the wording. "All consonants never come together" means we only exclude the single case where all 4 are in one block. If the question said "no two consonants are together", we would use the gap method.
Updated On: Feb 16, 2026
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Correct Answer: 576
Solution and Explanation
Step 1: Understanding the Concept:
The "never together" condition is usually handled by subtracting the cases where they all come together from the total permutations.
Step 2: Detailed Explanation:
The word 'VOWELS' has 6 letters: V, W, L, S (4 consonants) and O, E (2 vowels).
1. Total number of words:
Since all letters are distinct, Total = \(6! = 720\).
2. Cases where all consonants are together:
Treat {V, W, L, S} as one block. We then have 3 entities: {VWLS}, O, E.
Permutations of these 3 entities = \(3!\).
Internal permutations of the 4 consonants = \(4!\).
Together = \(3! \times 4! = 6 \times 24 = 144\).
3. Cases where all consonants never come together:
Result = Total - Together = \(720 - 144 = 576\).
Step 3: Final Answer:
The number of such words is 576.
The "never together" condition is usually handled by subtracting the cases where they all come together from the total permutations.
Step 2: Detailed Explanation:
The word 'VOWELS' has 6 letters: V, W, L, S (4 consonants) and O, E (2 vowels).
1. Total number of words:
Since all letters are distinct, Total = \(6! = 720\).
2. Cases where all consonants are together:
Treat {V, W, L, S} as one block. We then have 3 entities: {VWLS}, O, E.
Permutations of these 3 entities = \(3!\).
Internal permutations of the 4 consonants = \(4!\).
Together = \(3! \times 4! = 6 \times 24 = 144\).
3. Cases where all consonants never come together:
Result = Total - Together = \(720 - 144 = 576\).
Step 3: Final Answer:
The number of such words is 576.
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