If the letters of word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number

Write the six distinct letters of the word SACHIN in alphabetical (dictionary) order:
$$A \lt C \lt H \lt I \lt N \lt S$$

To find the serial (rank) of SACHIN, count all lexicographically smaller words that can be formed before it appears, position by position.

Step 1: Fix the first letter
Letters smaller than $$S$$ are $$A,C,H,I,N$$ (5 letters).
If any one of these 5 letters is kept first, the remaining 5 letters can be arranged in $$5! = 120$$ ways.
Hence words starting with a letter smaller than $$S$$: $$5 \times 120 = 600$$.

Step 2: First letter fixed as $$S$$, move to the second letter
Remaining letters: $$A,C,H,I,N$$ in alphabetical order.
The required word has $$A$$ in the second position. No letter among the remaining set is smaller than $$A$$, so no additional words are counted here.

Step 3: First two letters fixed as $$SA$$, move to the third letter
Remaining letters: $$C,H,I,N$$. The third letter of our word is $$C$$, which again is the smallest available. Additional words = 0.

Step 4: First three letters fixed as $$SAC$$, move to the fourth letter
Remaining letters: $$H,I,N$$. The next required letter is $$H$$ (smallest). Additional words = 0.

Step 5: First four letters fixed as $$SACH$$, move to the fifth letter
Remaining letters: $$I,N$$. The next letter is $$I$$ (smallest). Additional words = 0.

Step 6: First five letters fixed as $$SACHI$$
Only one letter $$N$$ remains, so no further counting is needed.

Total rank
Words before SACHIN = $$600$$.
Therefore, the dictionary position of SACHIN = $$600 + 1 = 601$$.

Option A which is: $$601$$