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

### Concept & Logic When a word has $n$ total letters but some letters are repeated, the total number of distinct arrangements is $n!$ divided by the factorial of the frequency of each repeated letter. $$P = \frac{n!}{p! \times q! \dots}$$ ### Step-by-Step Solution - **Given:** The word 'RIDDLED'. - Count total letters: $n = 7$. - Calculate the frequency of each letter: - R appears $1$ time. - I appears $1$ time. - D appears $3$ times. - L appears $1$ time. - E appears $1$ time. - Apply the formula to find the total ways = $\frac{7!}{3!}$. - Calculate: $\frac{5040}{6} = 840$. ### Exam Strategy & Shortcut Instead of computing $7!$ entirely and then dividing, expand out and cancel terms dynamically: $\frac{7 \times 6 \times 5 \times 4 \times 3!}{3!} = 7 \times 6 \times 5 \times 4 = 840$. ### Common Pitfall Forgetting to divide by the factorial of the repeated letters altogether, which would incorrectly yield an unadjusted $5040$. ### Final Answer Therefore, the correct answer is **840**.