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>
