Question - Determining the Lexicographic Rank of a Specific Word in Permutations

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:

1. Arrange the letters in alphabetical order: A, C, H, I, R, T

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:

For 'A': $5!$

For 'C': $5!$

For 'H': $5!$

For 'I': $5!$

3. Add these permutations to find the number of words before reaching the first word starting with 'R':

Total permutations before 'R': $4 \times 5!$

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:

For 'RAH', 'RAI', 'RAT': $3 \times 4!$

Total permutations after 'RA' and before 'RAC': $3 \times 4!$

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'.

6. Add the total permutations to get the rank of 'RACHIT':

Rank of RACHIT = $4 \times 5! + 3 \times 4! + 1$

7. Calculate factorials and add them up:

Rank of RACHIT = $4 \times 120 + 3 \times 24 + 1$

Rank of RACHIT = $480 + 72 + 1$

Rank of RACHIT = $553$

Therefore, the rank of the word 'RACHIT' is 553.