In how many different ways can the word "DAUGHTER" be arranged so that the vowels always come together?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 4320
When some letters must always come together, they are tied into one block and the block is arranged with the other letters as a single unit. The letters inside the block can then be arranged among themselves, and the two counts are multiplied.
The letters of DAUGHTER (8 different letters):
Vowels A, U, E (3)
Consonants D, G, H, T, R (5)
The arrangements (the vowels tied into one block):
Block (AUE) + 5 consonants \(=6\) units
The 6 units are arranged in \(6!=720\) ways
The 3 vowels inside the block are arranged in \(3!=6\) ways
Number of arrangements \(=720\times6=4320\)