Example Question - dictionary order
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Determining the Lexicographic Rank of a Specific Word in Permutations
<p>The word 'RACHIT' has 6 unique letters. To find the rank of the word 'RACHIT' when all the permutations of the letters are arranged in alphabetical order:</p> <p>1. Arrange the letters in alphabetical order: A, C, H, I, R, T</p> <p>2. Calculate the number of permutations that start with each letter before 'R' (the first letter of the given word), with the remaining 5 letters:</p> <p>For 'A': $5!$</p> <p>For 'C': $5!$</p> <p>For 'H': $5!$</p> <p>For 'I': $5!$</p> <p>3. Add these permutations to find the number of words before reaching the first word starting with 'R':</p> <p>Total permutations before 'R': $4 \times 5!$</p> <p>4. Now, calculate the number of permutations that start with 'RA', followed by each letter before 'C' (the second letter of 'RACHIT'), with the remaining 4 letters:</p> <p>For 'RAH', 'RAI', 'RAT': $3 \times 4!$</p> <p>Total permutations after 'RA' and before 'RAC': $3 \times 4!$</p> <p>5. The word 'RACHIT' is the next permutation following all those starting with 'RA' and not 'RAC'. Thus, it's the first permutation with the prefix 'RAC'.</p> <p>6. Add the total permutations to get the rank of 'RACHIT':</p> <p>Rank of RACHIT = $4 \times 5! + 3 \times 4! + 1$</p> <p>7. Calculate factorials and add them up:</p> <p>Rank of RACHIT = $4 \times 120 + 3 \times 24 + 1$</p> <p>Rank of RACHIT = $480 + 72 + 1$</p> <p>Rank of RACHIT = $553$</p> <p>Therefore, the rank of the word 'RACHIT' is 553.</p>
Ranking Permutations of a Given Word
<p>The word RACHIT contains 6 distinct letters.</p> <p>To find the rank of the word RACHIT when the letters are arranged in dictionary order:</p> <p>Step 1: Arrange the letters of the word in alphabetical order: A, C, H, I, R, T.</p> <p>Step 2: Count the number of words starting with each letter that is before R alphabetically and with the rest in any order.</p> <p>Step 3: For A as the first letter, we can arrange the remaining 5 letters in \(5!\) ways.</p> <p>Step 4: For C as the first letter, the count is again \(5!\) ways.</p> <p>Step 5: For H as the first letter, the count is \(5!\) ways.</p> <p>Step 6: For I as the first letter, the count is \(5!\) ways.</p> <p>Step 7: With R as the first letter, the next letter could be A, which gives \(4!\) arrangements for the remaining letters.</p> <p>Therefore, the rank of RACHIT = the sum of all the arrangements calculated above</p> <p>\(= 4 \times 5! + 1 \times 4!\)</p> <p>\(= 4 \times (5 \times 4 \times 3 \times 2 \times 1) + 4 \times 3 \times 2 \times 1\)</p> <p>\(= 4 \times 120 + 24\)</p> <p>\(= 480 + 24\)</p> <p>\(= 504\)</p> <p>So, the rank of the word RACHIT is 504 when the letters are arranged in dictionary order.</p>
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