In how many different ways can the letters of the word 'OFFICES' be arranged?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2250
Note: The given answer is as per the official NTA Answer Key. However, the answer given by NTA appears to be incorrect: the seven letters of ‘OFFICES’, with F repeated twice, can be arranged in 2520 ways, so Option (1) is the correct answer, as explained below. The letters of OFFICES can be shuffled in 2520 ways, but NTA’s key settled on a number no shuffle reaches.
When a word has repeated letters, swapping two identical letters does not give a new arrangement, so the count of all arrangements is divided by the factorial of each repeat: \(\displaystyle\frac{n!}{p!}\) for \(n\) letters with one letter repeated \(p\) times.
Letters: O, F, F, I, C, E, S — 7 letters, F twice
Arrangements \(=\displaystyle\frac{7!}{2!}=\frac{5040}{2}=2520\)
Option (1) 2520 — Should have been correct. It is \(\displaystyle\frac{7!}{2!}\), the count with the two F’s treated as identical.
Option (2) 2250 — Correct as per NTA; however, no counting of these letters gives 2250.