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In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together?

In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together? Correct Answer $$\frac{{4! \times 3!}}{{2!}}$$

ABACUS is a 6 letter word with 3 of the letters being vowels.If the 3 vowels have to appear together as stated in the question, then there will 3 consonants and a set of 3 vowels grouped together.One group of 3 vowels and 3 consonants are essentially 4 elements to be rearranged. The number of possible rearrangements is 4!The group of 3 vowels contains two a s and one uThe 3 vowels can rearrange amongst themselves in $$\frac{{3!}}{{2!}}$$ ways as the vowel a appears twice.Hence, the total number of rearrangements in which the vowels appear together are:$$\frac{{4! \times 3!}}{{2!}}$$

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