© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Interpret the Problem
We want to find the position of the word "THAMS" when all permutations of the letters in "MATHS" (namely M, A, T, H, and S) are arranged in alphabetical (dictionary) order.
Step 2: Determine the Alphabetical Order of Letters
The letters M, A, T, H, S in alphabetical order are A, H, M, S, T.
Step 3: Calculate the Permutations Starting with Letters Before 'T'
Before 'T', there are four letters: A, H, M, S. If any of these letters is the first letter, then the remaining 4 letters can be permuted in $4!$ ways.
$4! = 24$
Since there are 4 possible first letters (A, H, M, S) that come before 'T', the total permutations starting with these letters is:
$4 \times 4! = 4 \times 24 = 96.$
Hence, 96 words appear before any word beginning with 'T'.
Step 4: Count Words Starting with 'TA'
Within words starting with 'T', the next letter could be A, H, M, or S. All words beginning with 'TA' come before those beginning with 'TH'. For words that start with "TA", the remaining 3 letters can be arranged in $3!$ ways.
$3! = 6$
So, there are 6 words that begin with "TA...". Including these, the total so far is:
$96 + 6 = 102.$
Step 5: Positioning of "THAMS"
Next, consider words that begin with "TH". The remaining letters are A, M, and S. In alphabetical order, these letters are A, M, S. The possible permutations for these three letters are:
THAMS
THASM
THMAS
THMSA
THSAM
THSMA
"THAMS" is the first among these "TH..." words. It comes immediately after the 102 words we identified earlier, giving it position number:
$102 + 1 = 103.$
Step 6: Final Answer
The position (serial number) of the word "THAMS" is 103.
