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?

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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 2 A and 3 O, the...

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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 2 A and 3 O, the number of ways that can be done is `(6!)/(2!*3!)` = 60

Take all the vowels together as a single unit. Along with the 7 consonants there are 8 elements to be rearranged. This includes 2 M and 2 C. The number of possible arrangements here is `(8!)/(2!*2!)` = 10080.

The product of 10080 and 60 is 604800.

There are 604800 ways in which the letters of the word ACCOMMODATION can be arranged with the vowels together.

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