In how many ways can the letters of the word PUZZLE be arranged so that the vowels always come together?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 120
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. A letter that repeats \(k\) times means dividing by \(k!\).
PUZZLE: vowels U, E; consonants P, Z, Z, L (Z twice)
Units to arrange: [UE], P, Z, Z, L, that is 5 units with Z repeated
Arrangements of the units \(\displaystyle =\frac{5!}{2!}=\frac{120}{2}=60\)
U and E inside the block: \(2!=2\) ways
Total \(=60\times2=120\)
Why the other options are incorrect
Option (2) 280 — Incorrect. With U and E tied into one block there are only 120 arrangements, so 280 must include arrangements in which U and E are apart.
Option (3) 450 — Incorrect. Even all arrangements of PUZZLE are only \(6!\div2!=360\), so 450 is impossible.
Option (4) 630 — Incorrect. This is also more than the 360 arrangements of PUZZLE in all.