In how many different ways can the letters of the word OPERATE be arranged?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2520
\(n\) different letters can be arranged in \(n!\) ways. When a letter is repeated, swapping the copies gives no new word, so the total is divided by the factorial of the number of times each letter repeats: \(\frac{n!}{p!}\).
OPERATE has 7 letters: O, P, E, R, A, T, E, with E twice
Arrangements \(\displaystyle =\frac{7!}{2!}=\frac{5040}{2}=2520\)
Tip: 5040 in Option (3) is \(7!\), the count if every letter were different. Look for repeated letters before you multiply.