# 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 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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