In how many different ways can all the letters of the word COFFEE be arranged? (Each letter is used exactly once; the two F’s are identical and the two E’s are identical.)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 180
When letters repeat, swapping identical letters gives no new arrangement, so divide \(n!\) by the factorial of each repeat.
COFFEE has 6 letters: F twice, E twice, C and O once each
Arrangements \(\displaystyle=\frac{6!}{2!\times2!}=\frac{720}{4}=180\)
Why the other options are incorrect
Option (1) 360 — Incorrect. This is \(\frac{6!}{2!}\); it allows for only one of the two repeated letters.
Option (2) 90 — Incorrect. This divides by 2! once too often; only F and E repeat.
Option (4) 720 — Incorrect. This is \(6!\), which treats the repeated letters as different.