The set S=1,2,3,,12 is to be partitioned into three sets A, B, C of equal size.

Question

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

Accepted Answer

Each set has 4 elements. Choose A: $\binom{12}{4}$, then B from remaining 8: $\binom84$, then C is fixed: $\binom44$.

$$\binom{12}{4}\binom84\binom44 = \frac{12!}{4!8!}\cdot\frac{8!}{4!4!}\cdot\frac{4!}{4!0!} = \frac{12!}{(4!)^3}$$

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