JEE MAIN - Mathematics (2007 - No. 21)
The set S = {1, 2, 3, ........., 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
$${{12!} \over {{{(4!)}^3}}}\,\,$$
$${{12!} \over {{{(4!)}^4}}}\,\,$$
$${{12!} \over {3!\,\,{{(4!)}^3}}}$$
$${{12!} \over {3!\,\,{{(4!)}^4}}}$$
Explanation
The total number of ways is
$${}^{12}{C_4} \times {}^{12 - 4}{C_4} \times {}^{12 - 4 - 4}{C_4} = {}^{12}{C_4} \times {}^8{C_4} \times {}^4{C_4} = {{12!} \over {{{(4!)}^3}}}$$
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