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The set S=\{1,2,3, \ldots 12\} is to be partitioned into three sets \mathrm{A, B, C}  of equal size. Thus \mathrm{A \cup B \cup C=S . A \cap B=B \cap C=C \cap A=\phi}. The number of ways to partition \mathrm{S} is

Option: 1

\frac{12 !}{3 !(4 !)^{3}}


Option: 2

\frac{12 !}{3 !(3 !)^{3}}


Option: 3

\frac{12 !}{(4 !)^{3}}


Option: 4

\frac{12 !}{(3 !)^{4}}


Answers (1)

best_answer

Each of the three sets \mathrm{A, B, C} contains exactly 4 elements.

Thus, the number of ways of partitioning the set \mathrm{S} is

\frac{1}{3 !}\left({ }^{12} C_{4}\right)\left({ }^{8} C_{4}\right)\left({ }^{4} C_{4}\right)=\frac{1}{3 !} \frac{12 !}{4 ! 8 !} \times \frac{8 !}{4 ! 4 !}(1)=\frac{12 !}{3 !(4 !)^{3}}

Posted by

Divya Prakash Singh

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