1. **Problem:** How many different words can be formed with the letters of the word "MISSISSIPPI"? 2. **Step 1: Understand the problem** The word "MISSISSIPPI" has 11 letters with repetitions: M(1), I(4), S(4), P(2). 3. **Step 2: Formula for permutations with repeated letters** The number of distinct permutations of $n$ letters where there are repetitions is given by: $$\frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}$$ where $n_i!$ are factorials of the counts of each repeated letter. 4. **Step 3: Calculate total permutations** $$\text{Total} = \frac{11!}{1! \times 4! \times 4! \times 2!}$$ 5. **Step 4: Calculate permutations where all four 'I's come together** Treat the four 'I's as a single entity. Then letters are: M(1), S(4), P(2), I-block(1) = total 8 entities. Number of permutations: $$\frac{8!}{1! \times 4! \times 2!}$$ 6. **Step 5: Calculate permutations where four 'I's do NOT come together** $$\text{Not together} = \text{Total} - \text{All I's together} = \frac{11!}{1!4!4!2!} - \frac{8!}{1!4!2!}$$ 7. **Step 6: Compute values** $11! = 39916800$, $8! = 40320$, $4! = 24$, $2! = 2$ Calculate total: $$\frac{39916800}{1 \times 24 \times 24 \times 2} = \frac{39916800}{1152} = 34650$$ Calculate all I's together: $$\frac{40320}{1 \times 24 \times 2} = \frac{40320}{48} = 840$$ Calculate not together: $$34650 - 840 = 33810$$ **Final answers:** - Total different words: $34650$ - Words where four 'I's do not come together: $33810$