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

### Concept & Formula When multiple characters are repeated in a word, the total distinct permutations are calculated by dividing the total factorial by the product of the factorials of each character's frequency. $$P = \frac{n!}{p! \times q!}$$ ### Step-by-Step Solution - **Given:** The word 'RUMOUR'. - Count the total letters: $n = 6$. - Identify all duplicates: The letter 'R' appears $2$ times. The letter 'U' appears $2$ times. - Construct the formula: Total arrangements = $\frac{6!}{2! \times 2!}$. - Compute the numerator: $6! = 720$. - Compute the denominator: $2 \times 2 = 4$. - Divide: $\frac{720}{4} = 180$. ### Exam Strategy & Shortcut Knowing $6! = 720$, you just need to divide by $4$ since there are two pairs of identical letters. $720 / 4 = 180$. ### Common Pitfall Failing to spot one of the duplicate pairs (like missing the two 'U's) and dividing only by $2!$, which would give an incorrect answer of $360$. ### Final Answer Therefore, the correct answer is **180**.