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 $n_i$ are the counts of each repeated letter. 3. **Step 1: Identify letters and their counts in "science"** - Letters: s, c, i, e, n, c, e - Sorted letters: c, c, e, e, i, n, s - Counts: c=2, e=2, i=1, n=1, s=1 4. **Step 2: Calculate total permutations** - Total letters $n=7$ - Total permutations = $$\frac{7!}{2! \times 2!} = \frac{5040}{4} = 1260$$ 5. **Step 3: Find the position of "science"** We find how many permutations come before "science" alphabetically by fixing each letter from left to right and counting permutations of remaining letters. - First letter 's': letters before 's' are c, e, i, n - For each letter before 's', count permutations of remaining 6 letters. - For 'c' first: permutations = $$\frac{6!}{2! \times 2!} = \frac{720}{4} = 180$$ - For 'e' first: same count 180 - For 'i' first: same count 180 - For 'n' first: same count 180 - Total before 's' = $4 \times 180 = 720$ - Second letter 'c': letters before 'c' are none (since 'c' is smallest among remaining letters) - Add 0 - Third letter 'i': letters before 'i' among remaining letters c, e, e, i, n are c, e, e - For 'c' first: permutations of remaining 4 letters = $$\frac{4!}{2!} = \frac{24}{2} = 12$$ - For 'e' first: same 12, but two 'e's so count twice = 24 - Total before 'i' = 12 + 24 = 36 - Fourth letter 'e': letters before 'e' among remaining letters c, e, n are c - For 'c' first: permutations of remaining 3 letters = $$3! = 6$$ - Add 6 - Fifth letter 'n': letters before 'n' among remaining letters e, n are e - For 'e' first: permutations of remaining 2 letters = $$2! = 2$$ - Add 2 - Sixth letter 'c': letters before 'c' among remaining letters e, c are e - For 'e' first: permutations of remaining 1 letter = $$1! = 1$$ - Add 1 - Seventh letter 'e': last letter, no letters before - Add 0 6. **Step 4: Sum all counts and add 1 for the position of "science"** - Total before = 720 + 0 + 36 + 6 + 2 + 1 + 0 = 765 - Position = 765 + 1 = 766 **Final answer:** The word "science" is at position **766** among all its permutations arranged alphabetically.