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
$$\frac{{4! \times 3!}}{{2!}}$$
E
$$\frac{{5!}}{2}$$
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:$$\frac{{4! \times 3!}}{{2!}}$$