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The letters of the word ZENITH are writt...

The letters of the word ZENITH are written in all possible orders. How many words are possible if all these words are written out as in dictionary ? What is the rank of the word ZENITH?

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To solve the problem of finding the rank of the word "ZENITH" when all the letters are arranged in alphabetical order, we can follow these steps: ### Step-by-Step Solution: 1. **Identify the Letters in Alphabetical Order**: The letters in the word "ZENITH" are Z, E, N, I, T, H. Arranging them in alphabetical order gives us: - E, H, I, N, T, Z. 2. **Count Words Starting with Each Letter Before 'Z'**: We need to count how many words can be formed starting with each letter that comes before 'Z': - **Starting with E**: The remaining letters are H, I, N, T, Z. The number of arrangements is \(5! = 120\). - **Starting with H**: The remaining letters are E, I, N, T, Z. The number of arrangements is \(5! = 120\). - **Starting with I**: The remaining letters are E, H, N, T, Z. The number of arrangements is \(5! = 120\). - **Starting with N**: The remaining letters are E, H, I, T, Z. The number of arrangements is \(5! = 120\). - **Starting with T**: The remaining letters are E, H, I, N, Z. The number of arrangements is \(5! = 120\). Adding these gives us: \[ 120 + 120 + 120 + 120 + 120 = 600. \] 3. **Count Words Starting with 'Z'**: Now we consider words that start with 'Z'. We need to look at the letters that follow 'Z' in alphabetical order: - The remaining letters are E, H, I, N, T. 4. **Count Words Starting with 'ZE'**: - Remaining letters: H, I, N, T. The number of arrangements is \(4! = 24\). 5. **Count Words Starting with 'ZH'**: - Remaining letters: E, I, N, T. The number of arrangements is \(4! = 24\). 6. **Count Words Starting with 'ZI'**: - Remaining letters: E, H, N, T. The number of arrangements is \(4! = 24\). 7. **Count Words Starting with 'ZN'**: - Remaining letters: E, H, I, T. The number of arrangements is \(4! = 24\). 8. **Count Words Starting with 'ZT'**: - Remaining letters: E, H, I, N. The number of arrangements is \(4! = 24\). 9. **Count Words Starting with 'ZEN'**: - Remaining letters: H, I, T. The number of arrangements is \(3! = 6\). 10. **Count Words Starting with 'ZEH'**: - Remaining letters: I, N, T. The number of arrangements is \(3! = 6\). 11. **Count Words Starting with 'ZEI'**: - Remaining letters: H, N, T. The number of arrangements is \(3! = 6\). 12. **Count Words Starting with 'ZENI'**: - Remaining letters: H, T. The number of arrangements is \(2! = 2\). 13. **Count Words Starting with 'ZENITH'**: - This is the word we are interested in, so we stop here. ### Final Calculation of Rank: Now, we sum all the counts: \[ 600 \text{ (from letters before Z)} + 24 + 24 + 24 + 24 + 24 + 6 + 6 + 6 + 2 + 1 \text{ (for ZENITH itself)} = 616. \] Thus, the rank of the word "ZENITH" is **616**.
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