Rank of Word
Rank of a word is its position when all permutations of its letters are arranged in dictionary (alphabetical) order. For example, the word 'MOTHER' has a certain rank among all 6! = 720 permutations of its letters. Ranking requires systematically counting how many words come before the given word.
What You'll Learn
Introduction to Rank of Word
Rank of a word is its position when all permutations of its letters are arranged in dictionary (alphabetical) order. For example, the word 'MOTHER' has a certain rank among all 6! = 720 permutations of its letters. Ranking requires systematically counting how many words come before the given word.
Prerequisites
How to Solve Rank of Word Problems
Step 1: Write the word and sort its letters alphabetically
Step 2: Initialize rank = 1 (starting position)
Step 3: For each position from left to right:
Step 4: Count letters smaller than the current letter that are still available
Step 5: For each such smaller letter, calculate permutations of remaining letters
Step 6: Add these counts to rank
Step 7: Remove the current letter from available letters and move to next position
Example Problem
Example: Find the rank of the word 'MOTHER' among its permutations. Solution: Step 1: Alphabetical order of letters: E, H, M, O, R, T Step 2: Rank starts at 1 Step 3: Position 1: 'M' - letters before M: E, H (2 letters) - For E: remaining letters (H,M,O,R,T) = 5! = 120 - For H: remaining letters (E,M,O,R,T) = 5! = 120 - Add 240 to rank → rank = 241 Step 4: Remove M, remaining: E,H,O,R,T. Current letter: 'O' - Letters before O: E, H (2 letters) - For each: 4! = 24 → add 48 → rank = 289 Step 5: Remove O, remaining: E,H,R,T. Current letter: 'T' - Letters before T: E, H, R (3 letters) - For each: 3! = 6 → add 18 → rank = 307 Step 6: Remove T, remaining: E,H,R. Current letter: 'H' - Letters before H: E (1 letter) → 2! = 2 → add 2 → rank = 309 Step 7: Remove H, remaining: E,R. Current letter: 'E' - No letters before E → add 0 Step 8: Last letter: 'R' → rank = 309 Answer: 309th
Pro Tips & Tricks
- Sort the letters alphabetically before starting
- For repeated letters, divide by factorials of repeated letter counts
- Rank = 1 + sum of (number of permutations with smaller letter at each position)
- When a letter repeats, treat identical letters as indistinguishable
- For words with repeated letters, use n!/(p! × q! × ...) for permutations
- Double-check the count of smaller letters at each position
Shortcut Methods to Solve Faster
Common Mistakes to Avoid
Practice Worksheets
Practice makes perfect! Work through these worksheets to master Rank of Word. Each worksheet contains 20 questions with detailed explanations. Start from Worksheet 1 and progress through increasing difficulty levels.
Exam Importance
Rank of Word is an important topic for various competitive exams. Here's how frequently it appears:
Ready to Master Rank of Word?
Start with Worksheet 1 and work your way up to expert level! Each worksheet includes: