In how many ways can the letters of the word accommodation be arranged if all the letters are used and all the vowels are together?
There are 13 letters in the word ACCOMMODATION. The number of unique letters is 8, with 2 A, 2 C, 3 O and 2 M.
There are 6 vowels and 7 consonants. If the number of vowels is kept together, keeping in mind the...
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ACCOMMODATION is a thirteen letter word.
There are 6 vowels in it. Taking all the vowels together as a single unit, there would be 2 Cs, 2 Ms, 1D, 1 T and 1 N, i.e. 8 units altogether, including several repetitions.
There can be `(8!)/(2!*2!)` different arrangements possible with these words.
Again, among the 6 vowels there are 3 Os and 2 As. They can be arranged in `(6!)/(3!2!)` different ways.
Therefore, the number of ways the letters of the word “accommodation” can be arranged, if all the letters are used and all the vowels are together, is
`((8!)/(2!*2!))* ((6!)/(3!2!))= (8!*6!)/(3!*2!*2!*2!)=604800`