If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is

We need to find the serial number of the word THAMS when all permutations of the letters M, A, T, H, S are arranged in dictionary order.

The sorted order of letters is: A, H, M, S, T.

Words starting with letters before T:

Letters A, H, M, S come before T. Each gives $$4! = 24$$ words.

Count = $$4 \times 24 = 96$$

Words starting with T, then letters before H:

Remaining letters: {A, H, M, S}. Only A comes before H.

Count = $$1 \times 3! = 6$$

Words starting with TH, then letters before A:

Remaining letters: {A, M, S}. No letter comes before A.

Count = $$0$$

Words starting with THA, then letters before M:

Remaining letters: {M, S}. No letter comes before M.

Count = $$0$$

The word THAMS itself: Count = $$1$$

Serial number = $$96 + 6 + 0 + 0 + 1 = 103$$

The answer is Option A: $$103$$.