The set S={1,2,3,…12} is to be partitioned into three sets A, B, C of equal size. Thus A∪B∪C=S, A∩B=B∩C=C∩A=ϕ The number of ways to partition S is

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

A
$\frac{12!}{3! (4!)^{3}}$
B
$\frac{12!}{3! (3!)^{3}}$
C
$\frac{12!}{(4!)^{3}}$
D
$\frac{12!}{(3!)^{4}}$

Solution:

Each of the three sets $A, B, C$ contains exactly $4$ elements.
Thus, the number of ways of partitioning the set $S$ is

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

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