Step 1: Understanding the Question:
We need to find the rank of the word STOM in a dictionary formed by permutations of the letters M, O, S, T.
First, determine the alphabetical order of the letters:
M, O, S, T.
Note that the word MOST itself is formed by the letters in exact alphabetical order. Thus, MOST is the 1st word in the dictionary. The question asks for the rank counted from MOST, which is simply the rank of STOM.
Step 2: Computation Steps:
Total permutations = \( 4! = 24 \).
We count the number of words that appear before STOM.
1. Words starting with M:
First letter M fixed. Remaining 3 letters (O, S, T) can be arranged in \( 3! = 6 \) ways.
(These are words 1 to 6).
2. Words starting with O:
First letter O fixed. Remaining 3 letters (M, S, T) can be arranged in \( 3! = 6 \) ways.
(These are words 7 to 12).
3. Words starting with S:
The target word starts with S, so we look at the second letter.
Alphabetical order of remaining letters: M, O, T.
* Starts with SM:
Remaining letters (O, T) can be arranged in \( 2! = 2 \) ways.
(Words: SMOT, SMTO).
* Starts with SO:
Remaining letters (M, T) can be arranged in \( 2! = 2 \) ways.
(Words: SOMT, SOTM).
* Starts with ST:
The target word starts with ST. Check next letter.
Alphabetical order of remaining: M, O.
* Next letter M:
Word is STMO. This is the 1st word starting with ST.
* Next letter O:
Word is STOM. This is the 2nd word starting with ST.
Total Rank Calculation:
Rank = (Words starting with M) + (Words starting with O) + (Words starting with SM) + (Words starting with SO) + (Position of STOM in ST...)
Rank = \( 6 + 6 + 2 + 2 + 1 (\text{STMO}) + 1 (\text{STOM}) \)
Rank = \( 16 + 2 = 18 \).
Step 4: Required Answer:
The rank is 18.