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Question

The set S:{1,2,3,....,12} is to be partitioned into three sets A, B, C of equal size. Thus, ABC=S,AB=BC=AC=ϕ . The number of ways to partition S is

A
12!3!(4!)3
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B
12!3!(3!)4
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C
12!(4!)3
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D
12!(3!)4
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Solution

The correct option is C 12!(4!)3
There are 12 elements.

Three disjoint sets A,B and C of same size are to be formed.

Each set has 4 elements each.

No.of ways of selecting any 4 elements from 12 elements for set A=12C4

No.of ways of selecting any 4 elements from 8 elements for set B=8C4

No.of ways of selecting any 4 elements from 4 elements for set C=4C4

Total no. of partitions=12C4×8C4×4C4

=12!8!4!×8!4!4!×4!0!4!=12!(4!)3

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