In how many different ways can the letters of the word OPTICAL be arranged so that the vowels are never together?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 4320
OPTICAL has 7 different letters, of which O, I and A are vowels. “Never together” means that the three vowels never stand together as one block, so the arrangements wanted are all the arrangements minus those in which the three vowels come together.
All arrangements: \(7!=5040\)
Vowels together, taking OIA as one block: the block and P, T, C, L make 5 units, arranged in \(5!=120\) ways
The three vowels inside the block: \(3!=6\) ways, so \(120\times6=720\) arrangements have the vowels together
Vowels never together: \(5040-720=4320\)