1. Arrange the letters alphabetically

The word is MATHS. Its letters are: $M, A, T, H, S$

In alphabetical order: $A, H, M, S, T$

We need the dictionary rank of the word THAMS.

2. Count words before THAMS

We count how many valid permutations come before THAMS in dictionary order.

Step 1: First letter = T

Letters smaller than $T$ available are: $A, H, M, S$

For each choice, the remaining $4$ letters can be arranged in: $4! = 24$ ways.

So number of words before those starting with $T$ is: $4 \times 24 = 96$

Step 2: Fix first letter $T$, second letter = H

Remaining letters are: $A, H, M, S$ After fixing $T$, available letters are: $A, H, M, S$

Now second letter is $H$. Letters smaller than $H$ among available letters: $A$

For this choice, remaining $3$ letters can be arranged in: $3! = 6$ ways.

So add: $6$

Total so far: $96 + 6 = 102$

Step 3: Fix $TH$, third letter = A

Remaining letters are: $A, M, S$

The third letter is $A$. No letter smaller than $A$ is available. So add: $0$

Total remains: $102$

Step 4: Fix $THA$, fourth letter = M

Remaining letters are: $M, S$

The fourth letter is $M$. No letter smaller than $M$ is available. So add: $0$

Total remains: $102$

Step 5: Fix $THAM$, fifth letter = S

Only one letter remains, so add: $0$

3. Final rank

Dictionary rank is one more than the number of words before it: $102 + 1 = 103$

So the serial number of THAMS is: $\boxed{103}$

4. Compare with stored correct answer

Stored correct answer is A, which corresponds to $103$.

Hence, my derived answer agrees with the stored answer.