All the six letters of the name SACHIN are arranged to form different words without repeating any letter in any one word. The words so formed are then arranged as in a dictionary. What will be the position of the word SACHIN in that sequence?

The problem asks for the dictionary position of the word SACHIN when all permutations of its letters are arranged alphabetically. The letters in the name SACHIN are S, A, C, H, I, N. First, arrange these letters in alphabetical order: A, C, H, I, N, S. We calculate the number of words that come before SACHIN in the dictionary sequence. 1. Words starting with a letter alphabetically before 'S': The letters before 'S' are A, C, H, I, N (5 letters). For each of these 5 letters, if it is placed at the first position, the remaining 5 letters can be arranged in 5! ways. Number of words starting with A = 5! = 5 x 4 x 3 x 2 x 1 = 120. Number of words starting with C = 5! = 120. Number of words starting with H = 5! = 120. Number of words starting with I = 5! = 120. Number of words starting with N = 5! = 120. Total words starting with a letter before 'S' = 5 x 120 = 600. 2. Now, fix the first letter as 'S'. The remaining letters are A, C, H, I, N. The second letter of SACHIN is 'A'. Letters alphabetically before 'A' in the remaining set (A, C, H, I, N) = None (0 letters). So, 0 words start with 'S' followed by a letter before 'A'. (0 x 4! = 0). 3. Fix the first two letters as 'SA'. The remaining letters are C, H, I, N. The third letter of SACHIN is 'C'. Letters alphabetically before 'C' in the remaining set (C, H, I, N) = None (0 letters). So, 0 words start with 'SAC' followed by a letter before 'C'. (0 x 3! = 0). 4. Fix the first three letters as 'SAC'. The remaining letters are H, I, N. The fourth letter of SACHIN is 'H'. Letters alphabetically before 'H' in the remaining set (H, I, N) = None (0 letters). So, 0 words start with 'SACH' followed by a letter before 'H'. (0 x 2! = 0). 5. Fix the first four letters as 'SACH'. The remaining letters are I, N. The fifth letter of SACHIN is 'I'. Letters alphabetically before 'I' in the remaining set (I, N) = None (0 letters). So, 0 words start with 'SACHI' followed by a letter before 'I'. (0 x 1! = 0). 6. Fix the first five letters as 'SACHI'. The remaining letter is N. The sixth letter of SACHIN is 'N'. Letters alphabetically before 'N' in the remaining set (N) = None (0 letters). So, 0 words start with 'SACHIN' followed by a letter before 'N'. (0 x 0! = 0). Total number of words before SACHIN = 600 + 0 + 0 + 0 + 0 + 0 = 600. The position of the word SACHIN in the sequence is (Total words before SACHIN) + 1. Position = 600 + 1 = 601. Therefore, the position of the word SACHIN is 601. The final answer is C) 601.