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Question
The number of words not starting and ending with vowels formed using all the letters of the word 'UNIVERSITY' such that all the vowels are in alphabetical order is
The number of words not starting and ending with vowels formed using all the letters of the word 'UNIVERSITY' such that all the vowels are in alphabetical order is
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Solution
The correct option is A 8C4×6!
The word 'UNIVERSITY' has 4 vowels U, I, I, E and 6 consonants N, V, R, S, T, Y.
Since, no vowel appears at the starting and the ending. So, out of 6 consonants 2 come at the start and the end.This can be done in = 6C2×2! ways.
Now, the remaining 8 letters are placed at the 8 places in between the selected consonants. Here, 4 places where vowels are placed can be choosen in number of ways = 8C4.
The remaining 4 places where consonants are placed, are also choosen in the above process.
But, all the vowels in the alphabetical order E, I, I, U. So, their order is fixed. So, only the consonants will be arranged in number of ways=4!
Total number of ways=8C4×6!
The word 'UNIVERSITY' has 4 vowels U, I, I, E and 6 consonants N, V, R, S, T, Y.
Since, no vowel appears at the starting and the ending. So, out of 6 consonants 2 come at the start and the end.This can be done in = 6C2×2! ways.
Now, the remaining 8 letters are placed at the 8 places in between the selected consonants. Here, 4 places where vowels are placed can be choosen in number of ways = 8C4.
The remaining 4 places where consonants are placed, are also choosen in the above process.
But, all the vowels in the alphabetical order E, I, I, U. So, their order is fixed. So, only the consonants will be arranged in number of ways=4!
Total number of ways=8C4×6!
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