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

A. $\frac{6!}{2}$

B. 3! × 3!

C. $\frac{4!}{2}$

D. $\frac{4! \times 3!}{2!}$

Solution:

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 u
The 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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