# Moderate Permutation-Combination Solved QuestionAptitude Discussion

 Q. In how many ways can the letters of the word $ABACUS$ be rearranged such that the vowels always appear together?
 ✖ A. $\dfrac{6!}{2}$ ✖ B. $3! × 3!$ ✖ C. $\dfrac{4!}{2}$ ✔ D. $\dfrac{4! × 3!}{2!}$ ✖ E. $\dfrac{5!}{2}$

Solution:
Option(D) is correct

$ABACUS$ 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 $\dfrac{3!}{2!}$ ways as the vowel $a$ appears twice.

Hence, the total number of rearrangements in which the vowels appear together are:

$\dfrac{4! × 3!}{2!}$