<p>If the letters of words SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number</p>
Step-by-Step Solution
Key Concept: Count how many permutations come before SACHIN alphabetically by fixing each position and counting smaller letters. This uses the principle of lexicographic ordering combined with factorial counting.
<p><strong>Step 1:</strong> Arrange the letters of SACHIN in alphabetical order: A, C, H, I, N, S</p><p><strong>Step 2:</strong> Count permutations before SACHIN by fixing positions from left to right:</p><p><strong>Position 1 (S):</strong> Letters before S are A, C, H, I, N. Each gives 5! = 120 permutations. Total = 5 × 120 = 600</p><p><strong>Position 2 (A):</strong> S is fixed. Letters before A: none. Contributes 0</p><p><strong>Position 3 (C):</strong> S, A fixed. Letters before C: none (only H, I, N, C remain). Contributes 0</p><p><strong>Position 4 (H):</strong> S, A, C fixed. Letters before H: none (only I, N, H remain). Contributes 0</p><p><strong>Position 5 (I):</strong> S, A, C, H fixed. Letters before I: N. This gives 1 × 1! = 1 permutation (SACHIN → SACHIN with N before I means we'd get SACHNI). Contributes 1</p><p><strong>Position 6 (N):</strong> Only N left. Contributes 0</p><p><strong>Step 3:</strong> Total permutations before SACHIN = 600 + 0 + 0 + 0 + 1 + 0 = 601</p><p><strong>Step 4:</strong> Serial number of SACHIN = 601 + 1 = 602</p><p>∴ Answer: <strong>602</strong></p>
Correct Answer: A