In how many different ways can all the letters of the word BALLOON be arranged? (Each letter is used exactly once; the two L’s are identical and the two O’s are identical.)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1260
\(n\) letters can be arranged in \(n!\) ways if all are different. When a letter repeats, swapping its copies gives the same word, so the total is divided by the factorial of each repeat: \(\frac{n!}{p!\,q!}\).
BALLOON has 7 letters: B, A, N once each, L twice, O twice
Arrangements \(\displaystyle =\frac{7!}{2!\times2!}=\frac{5040}{4}=1260\)
Why the other options are incorrect
Option (2) 2520 — Incorrect. This is \(7!\div2!\); it allows for only one of the two repeated letters.
Option (3) 5040 — Incorrect. This is \(7!\), the count if all 7 letters were different.
Option (4) 630 — Incorrect. This divides by \(2!\) three times, but only two letters (L and O) repeat.