In how many different ways can all the letters of the word GARDEN be arranged? (Each letter is used exactly once.)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 720
\(n\) different letters, each used once, can be arranged in \(n!\) ways.
GARDEN has 6 letters, all different: G, A, R, D, E, N
Arrangements \(=6!=6\times5\times4\times3\times2\times1=720\)
Why the other options are incorrect
Option (1) 120 — Incorrect. This is \(5!\), the count for 5 letters; GARDEN has 6.
Option (2) 360 — Incorrect. This is \(\frac{6!}{2}\), as if one letter were repeated; no letter of GARDEN repeats.
Option (3) 5040 — Incorrect. This is \(7!\), the count for 7 letters.