All 7 letters of FORMULA, each used once, are written in a row. The three vowels O, U and A 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) → 720
Letters that must stay together are tied into one block. The block and the other letters are arranged, and then the letters inside the block are arranged.
Block [OUA] plus F, R, M and L: 5 units
The 5 units in a row: \(5!=120\) ways
Inside the block, O, U and A in \(3!=6\) orders
Rows \(=120\times6=720\)
Why the other options are incorrect
Option (1) 120 — Incorrect. This is \(5!\) only; it forgets the \(3!\) orders of the vowels inside the block.
Option (3) 4320 — Incorrect. This is \(6!\times3!\); with the 3 vowels as one block there are 5 units, not 6.
Option (4) 5040 — Incorrect. This is \(7!\), all the rows, with no condition.