Question

If all the letters of the word MOST are permuted and the words (with or without meaning) thus obtained are arranged in the dictionary order then the rank of the word STOM when counted from the rank of the word MOST, is

• A. 24
• B. 21
• C. 12
• D. 18

Hint:

Always write the letters in alphabetical order first. When fixing letters, if the fixed letter matches the target word's letter, move to the next position. If it comes before the target's letter alphabetically, add \( (\text{remaining})! \) to the count.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:

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: Detailed Explanation:

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: Final Answer:

The rank is 18.