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

### Concept & Permutations with Repetition When arranging a set of items where some items are identical, use the formula for permutations with indistinguishable objects: $$ \frac{n!}{p_1! \times p_2! \times \dots \times p_k!} $$ where $n$ is the total number of items, and $p_1, p_2$ are the frequencies of repeating items. ### Step-by-Step Solution 1. Count the total number of letters in "ENGINEERING": $n = 11$. 2. Identify and count the repeating letters: - E appears 3 times. - N appears 3 times. - G appears 2 times. - I appears 2 times. - R appears 1 time (does not affect division). 3. Apply the formula: $$ \frac{11!}{3! \times 3! \times 2! \times 2!} $$ 4. Expand and simplify: $$ \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 6 \times 2 \times 2} $$ Denominator equals $144$. $11! = 39,916,800$. $$ \frac{39916800}{144} = 277200 $$ ### Exam Strategy & Shortcut To simplify $11! / 144$ manually, do not multiply out $11!$. Cancel factors! The $6 \times 6$ in the denominator ($3! \times 3!$) cancels $36$ from the numerator (e.g., $9 \times 4$). The $2 \times 2 = 4$ cancels into the $8$ to leave $2$. Multiply the remaining smaller integers directly. ### Common Pitfall Missing a repeated letter. It is highly recommended to cross out letters as you tally their frequencies to ensure $p_1 + p_2 + \dots = n$. ### Final Answer Therefore, the correct answer is **277200**.