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Question
In how many ways can the letters of the word 'ALGEBRA' be arranged without changing the relative order of the vowels and consonants?
Solution
The relative positions of all the vowels and consonants is fixed.
The first letter is a vowel. It can be selected out of the 3 three vowels, of which two are same. So, the vowels can be arranged in selecting 3 things, of which two are of the same kind
⇒\[\frac{3!}{2!}\]
The second, third, fifth and sixth letters are consonants that can be filled by the available 4 consonants in 4! ways.
∴ By fundamental principle of counting, the number of words that can be formed = 4!\[\times \frac{3!}{2!}\]
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