1. Arrange the letters in alphabetical order

The word is SACHIN with letters: $S, A, C, H, I, N$

In dictionary order, we compare using alphabetical order: $A < C < H < I < N < S$

We need the rank of the word: $\text{SACHIN}$

2. Count words appearing before SACHIN

We calculate how many arrangements come before it lexicographically.

Position 1: S

Letters smaller than $S$ among $\{A,C,H,I,N,S\}$ are: $A, C, H, I, N$ There are $5$ such letters.

For each choice at the first position, the remaining $5$ letters can be arranged in: $5! = 120$ ways.

So number of words before those starting with $S$ is: $5 \times 120 = 600$

3. Position 2 onward Now fix first letter as $S$. Remaining letters are: $A, C, H, I, N$

The second letter in the word is $A$. Among remaining letters, no letter is smaller than $A$. So contribution is: $0$

Third letter is $C$. Remaining letters after fixing $S,A$ are: $C,H,I,N$ No letter is smaller than $C$ here. Contribution: $0$

Fourth letter is $H$. Remaining letters: $H,I,N$ No smaller letter than $H$. Contribution: $0$

Fifth letter is $I$. Remaining letters: $I,N$ No smaller letter than $I$. Contribution: $0$

Sixth letter is $N$, last remaining letter. Contribution: $0$

4. Find the rank

Number of words before SACHIN: $600$

Hence its serial number is: $600 + 1 = 601$

5. Option check

• A: $601$ ✅
• B: $600$ ❌
• C: $603$ ❌
• D: $602$ ❌

Therefore, the correct answer is: $\boxed{601}$