The set S = {1,2,3,…,12} is to be partitioned into three sets A,B, a

1.	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» 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

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»

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