In how many different ways can the letters of the word OFFICES be arranged?

### Concept & Logic When arranging words containing indistinguishable identical letters, we divide the total factorial of the word length by the factorial of the duplicates to eliminate overcounting. $$P = \frac{n!}{p! \times q!}$$ ### Step-by-Step Solution - **Given:** The word 'OFFICES'. - Count the total letters: $n = 7$. - Note the frequency of the letters: The letter 'F' appears $2$ times. All other letters (O, I, C, E, S) appear just $1$ time. - Set up the total number of valid arrangements = $\frac{7!}{2!}$. - Compute the numerator $7! = 5040$. - Divide by the duplicate factorial: $\frac{5040}{2} = 2520$. ### Exam Strategy & Shortcut With $7!$ evaluating to $5040$, spotting one pair of duplicates means you just quickly halve it: $5040 / 2 = 2520$. Always scan the word for multiple pairs, but here there is only one. ### Common Pitfall Failing to notice the double 'F' and hastily selecting $5040$ (option b). Carefully scan all given words for duplicate letters before solving. ### Final Answer Therefore, the correct answer is **2520**.