To find the serial number of the word "SACHIN" when arranged in all possible ways and written out as in a dictionary, we can use the concept of Permutations.

Permutations are the different ways in which a set of elements can be arranged. The formula to calculate the number of permutations of a set of n elements is given by n!.

In this case, we have a word "SACHIN" with 6 letters. So the total number of permutations of the word "SACHIN" is 6!.

Calculating the value of 6!:
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

So there are a total of 720 permutations of the word "SACHIN".

To determine the serial number of the word "SACHIN" in the alphabetical order when all permutations are arranged, we need to arrange all the permutations of the letters in alphabetical order and find the position of the word "SACHIN".

To do this, we can arrange the letters of the word "SACHIN" in alphabetical order. The letters in the word "SACHIN" are A, C, H, I, N, and S.

Arranging these letters in alphabetical order, we get the word "ACHINS".

Now, we need to find the serial number of the word "SACHIN" in the arranged alphabetical order.

To find the serial number, we need to consider all the words that come before the word "SACHIN" in alphabetical order.

We can start by considering the words starting with the letter 'A'. There are a total of 5! = 120 words starting with 'A'.

Next, we consider the words starting with the letter 'C'. There are a total of 5! = 120 words starting with 'C'.

Similarly, for the letters 'H', 'I', 'N', and 'S', there are 5! = 120 words each.

So, the total number of words that come before the word "SACHIN" in alphabetical order is:
120 + 120 + 120 + 120 + 120 + 120 = 720

Therefore, the word "SACHIN" appears at serial number 721 in the arranged alphabetical order.

Since the options given in the question are not in the correct format, we can conclude that none of the provided options is correct.