Key Concepts and Formulas

• Lexicographical Ordering: Arranging words or sequences in dictionary order, based on the alphabetical order of characters.
• Permutations: The number of ways to arrange n distinct objects is n! = n × (n-1) × (n-2) × ... × 2 × 1.
• Rank Determination: To find the rank of a word, count the number of words that precede it in lexicographical order, then add 1.

Step-by-Step Solution

First, let's analyze the given word and its letters: The word is SACHIN. The letters are S, A, C, H, I, N. All 6 letters are distinct.

Step 1: Arrange the letters alphabetically. To determine which words come before SACHIN, we first need to know the alphabetical order of its constituent letters. The letters in alphabetical order are: A, C, H, I, N, S.

Step 2: Count words starting with letters alphabetically smaller than the first letter of SACHIN. The first letter of SACHIN is 'S'. We need to count all the words that start with letters appearing before 'S' in our alphabetical list (A, C, H, I, N, S). The letters smaller than 'S' are A, C, H, I, N. There are 5 such letters.

• Words starting with 'A': If the first letter is fixed as 'A', the remaining 5 letters (C, H, I, N, S) can be arranged in the remaining 5 positions in $5!$ ways. $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ So, there are 120 words starting with 'A'.
• Words starting with 'C': Similarly, if the first letter is fixed as 'C', the remaining 5 letters (A, H, I, N, S) can be arranged in $5!$ ways. $5! = 120$ So, there are 120 words starting with 'C'.
• Words starting with 'H': If the first letter is fixed as 'H', the remaining 5 letters (A, C, I, N, S) can be arranged in $5!$ ways. $5! = 120$ So, there are 120 words starting with 'H'.
• Words starting with 'I': If the first letter is fixed as 'I', the remaining 5 letters (A, C, H, N, S) can be arranged in $5!$ ways. $5! = 120$ So, there are 120 words starting with 'I'.
• Words starting with 'N': If the first letter is fixed as 'N', the remaining 5 letters (A, C, H, I, S) can be arranged in $5!$ ways. $5! = 120$ So, there are 120 words starting with 'N'.

Total words starting with A, C, H, I, or N: $120 + 120 + 120 + 120 + 120 = 5 \times 120 = 600$. Why this step? All these 600 words will appear before any word starting with 'S' in dictionary order, and thus definitely before SACHIN.

Step 3: Consider words starting with 'S' and then the second letter. Since we have counted all words starting with letters before 'S', now we fix 'S' as the first letter. The target word is SACHIN. The first letter is 'S'. The remaining letters available are A, C, H, I, N. In alphabetical order, these are A, C, H, I, N. The second letter of SACHIN is 'A'. We need to count words starting with 'S' followed by a letter smaller than 'A' from the remaining available letters (C, H, I, N). Since 'A' is the smallest of the remaining letters, there are no letters smaller than 'A'. So we need to count words starting with 'S' and a letter smaller than 'A' = 0. Why this step? We are systematically building up the word SACHIN. After fixing 'S', we check the next position. If the second letter of SACHIN is 'A', we must account for any words starting with 'S' followed by a letter alphabetically smaller than 'A' using the remaining letters.

Step 4: Consider words starting with 'SA' and then the third letter. The first two letters are fixed as 'SA'. The remaining letters available are C, H, I, N. In alphabetical order, these are C, H, I, N. The third letter of SACHIN is 'C'. We need to count words starting with 'SA' followed by a letter smaller than 'C' from the remaining available letters (H, I, N). There are no letters smaller than 'C' from the set {H,I,N}. So, number of words starting with 'SA' and a letter before 'C' = 0. Why this step? We continue the process. After fixing 'SA', we check the third position. If the third letter of SACHIN is 'C', we must account for any words starting with 'SA' followed by a letter alphabetically smaller than 'C' using the remaining letters.

Step 5: Consider words starting with 'SAC' and then the fourth letter. The first three letters are fixed as 'SAC'. The remaining letters available are H, I, N. In alphabetical order, these are H, I, N. The fourth letter of SACHIN is 'H'. We need to count words starting with 'SAC' followed by a letter smaller than 'H' from the remaining available letters (I, N). There are no letters smaller than 'H' from the set {I,N}. So, number of words starting with 'SAC' and a letter before 'H' = 0. Why this step? Same reasoning as above, extending to the fourth position.

Step 6: Consider words starting with 'SACH' and then the fifth letter. The first four letters are fixed as 'SACH'. The remaining letters available are I, N. In alphabetical order, these are I, N. The fifth letter of SACHIN is 'I'. We need to count words starting with 'SACH' followed by a letter smaller than 'I' from the remaining available letters (N). There are no letters smaller than 'I' from the set {N}. So, number of words starting with 'SACH' and a letter before 'I' = 0. Why this step? Same reasoning, extending to the fifth position.

Step 7: Consider words starting with 'SACHI' and then the sixth letter. The first five letters are fixed as 'SACHI'. The remaining letter available is N. In alphabetical order, this is N. The sixth letter of SACHIN is 'N'. We need to count words starting with 'SACHI' followed by a letter smaller than 'N' from the remaining available letter (none). There are no letters smaller than 'N' from the remaining letters. So, number of words starting with 'SACHI' and a letter before 'N' = 0. Why this step? Same reasoning, extending to the sixth position.

Step 8: Determine the rank of SACHIN. All the words we counted in the preceding steps come before SACHIN. Total number of words before SACHIN = 600 (from Step 2) + 0 (from Step 3) + 0 (from Step 4) + 0 (from Step 5) + 0 (from Step 6) + 0 (from Step 7) = 600. The rank of the word SACHIN is (Number of words before SACHIN) + 1. Rank of SACHIN = $600 + 1 = 601$.

Common Mistakes & Tips

• Forgetting to add 1: The count of words before the target word is not its rank. You must add 1 to get the actual serial number.
• Incorrectly determining "smaller" letters: Always ensure you are comparing the current letter of the target word with the alphabetically smaller letters from the set of remaining, unused letters.

Summary

To determine the rank of the word SACHIN, we systematically counted all words that would appear before it in a dictionary. We first found all words starting with letters alphabetically prior to 'S'. Then, we fixed 'S' and proceeded to compare the subsequent letters of SACHIN, counting the words that could be formed with lexicographically smaller letters in each position. Finally, we added 1 to the total count to find the rank of SACHIN.

Final Answer:

The final answer is \boxed{601}, which corresponds to option (A).