All 8 letters of TRIANGLE, each used once, are written in a row. The three vowels I, A and E must stand side by side as one group, in any order among themselves. How many such rows are possible?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 4320
Letters that must stay together are tied into one block; the block is arranged with the other letters, and then the letters inside the block are arranged.
TRIANGLE has 8 letters, all different; vowels: I, A, E; consonants: T, R, N, G, L
Block (IAE) + 5 consonants = 6 units, arranged in \(6!=720\) ways
Inside the block, the 3 vowels are arranged in \(3!=6\) ways
Arrangements \(=720\times6=4320\)
Why the other options are incorrect
Option (1) 720 — Incorrect. This is only \(6!\); it leaves out the \(3!\) orders of the vowels inside the block.
Option (3) 40320 — Incorrect. This is \(8!\), all the arrangements, including those where the vowels are apart.
Option (4) 1440 — Incorrect. This is \(6!\times2!\), as if the block held only 2 letters; it holds 3 vowels.