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

### Concept & Logic To find distinct linear permutations of a multiset (a word with repeating characters), take the factorial of the total letter count and divide by the factorials of the counts of any repeated letters. $$P = \frac{n!}{p!}$$ ### Step-by-Step Solution - **Given:** The word 'INCREASE'. - Count the total letters: $n = 8$. - Identify duplicates: The letter 'E' appears $2$ times. All other characters (I, N, C, R, A, S) appear once. - Set up the calculation: Total arrangements = $\frac{8!}{2!}$. - Recall or calculate $8! = 40320$. - Divide by $2$: $\frac{40320}{2} = 20160$. ### Exam Strategy & Shortcut If you know $7! = 5040$, you can quickly calculate $8!$ as $5040 \times 8 = 40320$. Spotting the single pair of identical letters means you just take half of $40320$ to get $20160$. ### Common Pitfall Miscounting the length of the word (e.g., counting 7 letters instead of 8) or ignoring the repeated 'E's, which would lead directly to the unadjusted trap answer of $40320$ (option a). ### Final Answer Therefore, the correct answer is **20160**.