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Question

How many different words can be formed by using all the letters of the word ALLAHABAD?In how many of them vowels occupy the even positions?

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Solution

There are 9 letters in the word ALLAHABAD out of which 4 are As,2 are Ls and the rest are all distinct.
So, the requisite number of words=9!4!2!=9×8×7×6×5×4!4!×2=9×4×7×6×5=7560
There are 4 vowels and all are alike i.e. 4As. Also, there are 4 even places viz 2nd,4th,6th and 8th. So, these 4 even places can be occupied by 4 vowels in 4!4!=1 way. Now, we are left with 5 places in which 5 letters, of which two are alike (2Ls) and other distinct, can be arranged in 5!2! ways.

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