# 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. $$\frac{{5!}}{2}$$

**Answer: Option D **

__Solution(By Examveda Team)__

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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A. 7560,60,1680

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C. 7650,200,4444

D. None of these

A. 8 × 9!

B. 8 × 8!

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D. 9 × 8!

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