In how many different ways can the letters of the word 'HARDWARE' be arranged in such a way that the vowels always come together?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 1080
To keep the vowels together, tie them into one block and arrange the block with the other letters; then arrange the vowels inside the block. Letters that repeat are divided out, since swapping two identical letters gives no new word.
HARDWARE has 8 letters: vowels A, A, E (3) and consonants H, R, D, W, R (5, with R twice)
5 consonants + 1 vowel block \(=6\) units, with R repeated: \(\displaystyle\frac{6!}{2!}=\frac{720}{2}=360\)
Vowels inside the block, with A repeated: \(\displaystyle\frac{3!}{2!}=3\)
Total \(=360\times3=1080\)