Permutations & Combinations
Arrangement of books
Grade 11
Question:
<p>There are three copies each of four different books. The number of ways in which they can be arranged on a shelf is</p>
<p>\(\dfrac{12!}{(3!)^4}\)</p>
<p>\(\dfrac{12!}{(4!)^3}\)</p>
<p>\(\dfrac{21!}{(3!)^4 4!}\)</p>
<p>\(\dfrac{12!}{(4!)^3 3!}\)</p>
Step-by-Step Solution
Key Concept: When arranging identical objects of different types, use the multinomial coefficient formula: n!/(n₁!×n₂!×...×nₖ!), where n is total items and nᵢ are frequencies of each identical group.
<p><strong>Step 1:</strong> Identify the total number of books: 3 copies × 4 different books = 12 books total</p><p><strong>Step 2:</strong> Recognize that books of the same type are identical (indistinguishable from each other)</p><p><strong>Step 3:</strong> Apply multinomial coefficient formula for arrangements with identical objects:</p><p>Number of arrangements = 12!/(3! × 3! × 3! × 3!)</p><p><strong>Step 4:</strong> Calculate:</p><p>= 12!/(3!)⁴</p><p>= 479001600/1296</p><p>= 369600</p><p>∴ Answer: A</p>
Correct Answer: A