Key Concepts and Formulas

• Lexicographical Ordering: Arrangement of words in dictionary order (alphabetical order).
• Permutations: The number of ways to arrange n distinct objects is n! = n × ( n - 1) × ... × 2 × 1.
• Fundamental Principle of Counting: If there are m ways to do one thing and n ways to do another, then there are m × n ways to do both.

Step-by-Step Solution

Step 1: Arrange the Letters Alphabetically

First, we need to arrange the letters of the word "MATHS" in alphabetical order. This will help us determine which words come before "THAMS" in the dictionary.

The letters are A, H, M, S, T.

Step 2: Count Words Starting with Letters Before 'T'

We want to find the number of words that would appear before any word starting with 'T'. These are words starting with 'A', 'H', 'M', or 'S'.

• Words starting with 'A': If 'A' is fixed as the first letter, the remaining 4 letters (H, M, S, T) can be arranged in the remaining 4 positions in 4! ways. Number of words starting with 'A' = $4! = 4 \times 3 \times 2 \times 1 = 24$. We use the factorial because there are 4 options for the second letter, 3 for the third, 2 for the fourth, and 1 for the last.

• Words starting with 'H': Similarly, if 'H' is fixed as the first letter, the remaining 4 letters (A, M, S, T) can be arranged in 4! ways. Number of words starting with 'H' = $4! = 24$.

• Words starting with 'M': If 'M' is fixed as the first letter, the remaining 4 letters (A, H, S, T) can be arranged in 4! ways. Number of words starting with 'M' = $4! = 24$.

• Words starting with 'S': If 'S' is fixed as the first letter, the remaining 4 letters (A, H, M, T) can be arranged in 4! ways. Number of words starting with 'S' = $4! = 24$.

Total number of words starting with letters alphabetically before 'T' = $24 + 24 + 24 + 24 = 4 \times 24 = 96$.

Step 3: Count Words Starting with 'T' and the Second Letter Before 'H'

Now we consider words starting with 'T', since our target word "THAMS" starts with 'T'. The remaining letters available are A, H, M, S. We need to count words starting with 'T' whose second letter comes alphabetically before 'H'.

From {A, H, M, S}, the only letter before 'H' is 'A'.

• Words starting with 'TA': If 'TA' are fixed as the first two letters, the remaining 3 letters (H, M, S) can be arranged in the remaining 3 positions in 3! ways. Number of words starting with 'TA' = $3! = 3 \times 2 \times 1 = 6$.

Step 4: Count Words Starting with 'TH' and the Third Letter Before 'A'

The target word is "THAMS". We have accounted for words before 'T' and words starting with 'TA'. Now we consider words starting with 'TH'. The remaining letters available are A, M, S. The third letter of our target word is 'A'. We need to count words starting with 'TH' whose third letter comes alphabetically before 'A'.

From {A, M, S}, there are no letters that come alphabetically before 'A'. Therefore, the number of words starting with 'TH' and a third letter before 'A' is 0.

Step 5: Count Words Starting with 'THA' and the Fourth Letter Before 'M'

The target word is "THAMS". We are now considering words starting with 'THA'. The remaining letters available are M, S. The fourth letter of our target word is 'M'. We need to count words starting with 'THA' whose fourth letter comes alphabetically before 'M'.

From {M, S}, there are no letters that come alphabetically before 'M'. Therefore, the number of words starting with 'THA' and a fourth letter before 'M' is 0.

Step 6: Count Words Starting with 'THAM' and the Fifth Letter Before 'S'

The target word is "THAMS". We are now considering words starting with 'THAM'. The remaining letter available is S. The fifth letter of our target word is 'S'. We need to count words starting with 'THAM' whose fifth letter comes alphabetically before 'S'.

From {S}, there are no letters that come alphabetically before 'S'. Therefore, the number of words starting with 'THAM' and a fifth letter before 'S' is 0.

Step 7: Calculate the Total Number of Words Before "THAMS"

Summing up the counts from the previous steps: Total words before "THAMS" = (Words starting with A, H, M, S) + (Words starting with TA) Total words before "THAMS" = $96 + 6 = 102$.

Step 8: Determine the Rank of "THAMS"

Since there are 102 words that come before "THAMS" in the dictionary arrangement, the word "THAMS" itself will be the 103rd word. Rank of "THAMS" = (Total words before "THAMS") + 1 Rank of "THAMS" = $102 + 1 = 103$.

Common Mistakes & Tips:

• Alphabetical Order: Ensure the initial ordering is correct. Double-check it.
• Adding 1: Don't forget to add 1 to the count of preceding words to get the rank.

Summary

To find the rank of "THAMS" in the lexicographically ordered permutations of "MATHS", we counted the number of words preceding it. We considered words starting with letters before 'T', then words starting with 'T' and a second letter before 'H', and so on. Summing these counts and adding 1 gives the rank of "THAMS", which is 103.

Final Answer

The final answer is $\boxed{103}$, which corresponds to option (A).