Question

The set S = {1,2,3,…,12} is to be partitioned into three sets A,B, and C of equal sizes. A ⋃ B ⋃ C = S, A ⋂ B = B ⋂ C = C ⋂ A = Φ Find the number of ways to partitions.

Answer 1

There are 12 elements. 

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

⇒ Each set has 4 elements. 

Number of ways of selecting any 4 elements from 12 elements for set A = ¹²C₄ 

Number of ways of selecting any 4 elements from 8 elements for set B = ⁸C₄ 

Number of ways of selecting any 4 elements from 4 elements for C = ⁴C₄ 

∴ Total number of partitions = ¹²C₄ × ⁸C₄ × ⁴C₄