Permutations & Combinations
Rank of a Word
Grade 11
Question:
<p>Find the rank of the word MOTHER when all permutations of the letters M, O, T, H, E, R are arranged in alphabetical order.</p>
Step-by-Step Solution
Key Concept: Systematically count permutations before the target word by fixing prefixes and arranging remaining letters in alphabetical order.
<p><strong>Step 1:</strong> Words starting with E = 5! = 120<br/>Words starting with H = 5! = 120</p><p><strong>Step 2:</strong> Words starting with M but before MO:</p><p>Words starting with ME = 4! = 24<br/>Words starting with MH = 4! = 24</p><p><strong>Step 3:</strong> Words starting with MO but before MOT:</p><p>Words starting with MOE = 3! = 6<br/>Words starting with MOH = 3! = 6<br/>Words starting with MOR = 3! = 6</p><p><strong>Step 4:</strong> Words starting with MOT but before MOTH:</p><p>Words starting with MOTE = 2! = 2</p><p><strong>Step 5:</strong> Words starting with MOTH but before MOTHE:</p><p>Words starting with MOTHER = 1</p><p><strong>Step 6:</strong> Rank = 2(120) + 2(24) + 3(6) + 2 + 1 = 240 + 48 + 18 + 2 + 1 = 309</p>
Correct Answer: 309