1. **Problem Statement:** How many distinct ways can the letters of the word ENGINEERING be arranged? 2. **Understanding the problem:** The word ENGINEERING has 11 letters in total. We need to find the number of distinct permutations considering repeated letters. 3. **Count the frequency of each letter:** - E appears 3 times - N appears 3 times - G appears 2 times - I appears 2 times - R appears 1 time 4. **Formula for permutations with repeated letters:** $$\text{Number of distinct permutations} = \frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}$$ where $n$ is total letters, and $n_1, n_2, ..., n_k$ are frequencies of each repeated letter. 5. **Apply the formula:** $$\frac{11!}{3! \times 3! \times 2! \times 2! \times 1!} = \frac{11!}{3!3!2!2!}$$ 6. **Calculate factorial values:** - $11! = 39916800$ - $3! = 6$ - $2! = 2$ 7. **Calculate denominator:** $$3! \times 3! \times 2! \times 2! = 6 \times 6 \times 2 \times 2 = 144$$ 8. **Final calculation:** $$\frac{39916800}{144} = 277200$$ **Answer:** The letters of ENGINEERING can be arranged in **277200** distinct ways. This matches option (a) which is $$\frac{11!}{3!3!2!2!}$$. **Alternative approach:** Instead of calculating factorials directly, you can think stepwise: - Choose positions for 3 E's out of 11: $\binom{11}{3}$ - Then choose positions for 3 N's out of remaining 8: $\binom{8}{3}$ - Then choose positions for 2 G's out of remaining 5: $\binom{5}{2}$ - Then choose positions for 2 I's out of remaining 3: $\binom{3}{2}$ - Remaining 1 position for R Multiply all: $$\binom{11}{3} \times \binom{8}{3} \times \binom{5}{2} \times \binom{3}{2} \times 1 = 165 \times 56 \times 10 \times 3 = 277200$$ This confirms the same answer.