Question

The set `S""=""{1,""2,""3,"" ,""12)` is to be partitioned into three sets A, B, C of equal size. Thus, `AuuBuuC""=""S ,""AnnB""=""BnnC""=""AnnC""=varphi` . The number of ways to partition S is (1) `(12 !)/(3!(4!)^3)` (2) `(12 !)/(3!(3!)^4)` (3) `(12 !)/((4!)^3)` (4) `(12 !)/((4!)^4)`
A. `(12!)/((4!)^(3))`
B. `(12!)/((3!)^(4))`
C. `(12!)/(3!(4!)^(3))`
D. `(12!)/(3!(3!)^(4))`

Answer 1

The number of ways of partitioning S into three disjoint sets is same as the number of ways of distributing 12 distinct items to three persons each getting same number of items.
`:.` Required number of ways `=""^(12)C_(4)xx""^(8)C_(4)xx""^(4)C_(4)=(12!)/((4!)^(3))`