Permutations & Combinations
Rank of a Word
Grade 11
Question:
<p>Find the rank of the word KRISNA when all permutations of the letters K, R, I, S, N, A are arranged in alphabetical order.</p>
Step-by-Step Solution
Key Concept: Count all permutations that come alphabetically before the given word by systematically considering prefixes.
<p><strong>Step 1:</strong> Count words starting with letters before K in alphabetical order:</p><p>Words starting with A = 5! = 120<br/>Words starting with I = 5! = 120</p><p><strong>Step 2:</strong> Count words starting with K but before KR:</p><p>Words starting with KA = 4! = 24<br/>Words starting with KI = 4! = 24<br/>Words starting with KN = 4! = 24</p><p><strong>Step 3:</strong> Count words starting with KR but before KRI:</p><p>Words starting with KRA = 3! = 6</p><p><strong>Step 4:</strong> Count words starting with KRI but before KRIS:</p><p>Words starting with KRIA = 2! = 2<br/>Words starting with KRIN = 2! = 2</p><p><strong>Step 5:</strong> Count words starting with KRIS but before KRISNA:</p><p>Words starting with KRISA = 1! = 1</p><p><strong>Step 6:</strong> KRISNA itself = 1</p><p><strong>Step 7:</strong> Rank = 2(120) + 3(24) + 6 + 2(2) + 2(1) + 1 = 240 + 72 + 6 + 4 + 2 + 1 = 324</p>
Correct Answer: 324