1. **Problem statement:** Find the position of the word "science" among all its permutations when arranged in alphabetical order. 2. **Formula and rules:** The total number of permutations of a word with repeated letters is given by $$\frac{n!}{n_1! \times n_2! \times \cdots}$$ where $n$ is the total number of letters and $n_1, n_2, \ldots$ are the counts of each repeated letter. 3. **Step 1: Identify letters and their counts in "science":** The letters are s, c, i, e, n, c, e. - Total letters $n=7$ - Counts: c appears 2 times, e appears 2 times, s, i, n appear once each. 4. **Step 2: List letters in alphabetical order:** c, e, i, n, s 5. **Step 3: Calculate position by counting permutations starting with letters alphabetically before each letter in the word:** - First letter: s - Letters before s: c, e, i, n - For each letter before s, count permutations of remaining 6 letters. 6. **Step 4: Calculate permutations for each case:** - Total permutations of 7 letters with repeats: $$\frac{7!}{2! \times 2!} = \frac{5040}{4} = 1260$$ - For first letter c (before s): permutations of remaining letters (6 letters with c and e repeats): $$\frac{6!}{1! \times 2!} = \frac{720}{2} = 360$$ - For first letter e: same calculation, 360 permutations - For first letter i: no repeats left, so permutations of 6 letters with c and e repeats: $$\frac{6!}{2! \times 1!} = \frac{720}{2} = 360$$ - For first letter n: same as above, 360 permutations 7. **Step 5: Sum permutations before words starting with s:** $$360 + 360 + 360 + 360 = 1440$$ 8. **Step 6: Now fix first letter s and move to second letter:** - Word is s c i e n c e - Second letter is c - Letters available for second position: c, e, i, n, e - Letters before c: none - So no permutations added here. 9. **Step 7: Third letter is i** - Letters left: e, i, n, e - Letters before i: e - Count permutations starting with e at third position: Remaining letters: e, n, e Number of permutations: $$\frac{3!}{2!} = \frac{6}{2} = 3$$ 10. **Step 8: Add 3 permutations before words starting with s c i** 11. **Step 9: Fourth letter is e** - Letters left: e, n, e - Letters before e: none - No permutations added. 12. **Step 10: Fifth letter is n** - Letters left: e, e - Letters before n: e - Count permutations starting with e at fifth position: Remaining letters: e Number of permutations: 1 13. **Step 11: Add 1 permutation before words starting with s c i e n** 14. **Step 12: Sixth letter is c** - Letters left: e - Letters before c: none - No permutations added. 15. **Step 13: Seventh letter is e** - Last letter, no permutations left. 16. **Step 14: Calculate total permutations before the word "science":** $$1440 + 3 + 1 = 1444$$ 17. **Step 15: Position of "science" is one more than permutations before it:** $$1444 + 1 = 1445$$ **Final answer:** The word "science" is at position **1445** among all its permutations arranged alphabetically.