<p>The set \(S = \{1, 2, 3, \ldots, 12\}\) is to be partitioned into three sets \(A\), \(B\), \(C\) of equal size. Thus, \(A \cup B \cup C = S\), \(A \cap B = B \cap C = A \cap C = \phi\). The number of ways to partition \(S\) is</p>
<p>\(\dfrac{12!}{3!(4!)^3}\)</p>
<p>\(\dfrac{12!}{3!(3!)^4}\)</p>
<p>\(\dfrac{12!}{(4!)^3}\)</p>
<p>\(\dfrac{12!}{(3!)^4}\)</p>
Step-by-Step Solution
Key Concept: When partitioning a set into identical groups, you must divide by the number of ways to arrange those groups since the partition {A,B,C} is the same as {B,A,C}. The formula is C(12,4)×C(8,4)×C(4,4) divided by 3! to account for the indistinguishability of the three sets.
<p><strong>Step 1:</strong> Each set A, B, C must contain exactly 12÷3 = 4 elements.</p><p><strong>Step 2:</strong> If sets were distinguishable (ordered), we'd choose 4 elements for A from 12, then 4 from remaining 8 for B, then 4 for C: C(12,4)×C(8,4)×C(4,4)</p><p><strong>Step 3:</strong> Calculate: C(12,4) = 495, C(8,4) = 70, C(4,4) = 1</p><p>Product = 495 × 70 × 1 = 34,650</p><p><strong>Step 4:</strong> Since A, B, C are indistinguishable in a partition (the order doesn't matter), divide by 3! = 6 to avoid overcounting.</p><p><strong>Step 5:</strong> Number of partitions = 34,650 ÷ 6 = 5,775</p><p>∴ Answer: C (which equals <strong>12!/(4!×4!×4!×3!)</strong> or <strong>5,775</strong>)</p>
Correct Answer: C