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Step-by-Step Solution
Step 1: List the letters in alphabetical order
The letters in "SACHIN" are: S, A, C, H, I, N.
In alphabetical order, they become A, C, H, I, N, S.
Step 2: Count words starting with letters that come before "S"
In the alphabetical list (A, C, H, I, N, S), the letters A, C, H, I, and N all come before S.
If a letter other than S is at the first position, the remaining 5 letters can be arranged in $5!$ ways.
Since there are 5 such letters that could come first, the number of arrangements before any word starting with S is
$$
5 \times 5! = 5 \times 120 = 600.
$$
Step 3: Fix "S" and move to the second letter "A"
After choosing S for the first position, the remaining letters are A, C, H, I, N.
Among these, "A" is the smallest, so there are no letters that come before A. Hence,
$$
0 \times 4! = 0.
$$
Step 4: Fix "SA" and move to the third letter "C"
Now the remaining letters are C, H, I, N. "C" is the smallest in the set, so there are no letters before it.
Thus,
$$
0 \times 3! = 0.
$$
Step 5: Fix "SAC" and move to the fourth letter "H"
The remaining letters are H, I, N. "H" is the smallest in this set, so again,
$$
0 \times 2! = 0.
$$
Step 6: Fix "SACH" and move to the fifth letter "I"
The remaining letters are I, N. "I" is the smallest, so
$$
0 \times 1! = 0.
$$
Step 7: Combine all permutations counted before "SACHIN"
By summing all the permutations so far, we get
$$
600 + 0 + 0 + 0 + 0 = 600.
$$
Therefore, the rank of "SACHIN" is
$$
600 + 1 = 601.
$$
The extra 1 is because we start counting from the very next position after all the permutations before it.
Answer
The word "SACHIN" appears at the serial number 601.
